Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.8% → 88.4%

Time: 18.6s

Alternatives: 15

Speedup: 32.4×

• 27.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \ell \cdot \cos k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.2 \cdot 10^{+86}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(t\_m \cdot \left(\frac{2 \cdot \left(t\_m \cdot \left(t\_m \cdot \sin k\right)\right)}{t\_2} + \left(k \cdot k\right) \cdot \frac{\sin k}{t\_2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m}}{\sin k \cdot \left(\left(2 + \frac{k}{t\_m \cdot \frac{t\_m}{k}}\right) \cdot \left(\tan k \cdot \frac{t\_m}{\ell}\right)\right)}}{\frac{t\_m}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* l (cos k))))
   (*
    t_s
    (if (<= t_m 9.2e+86)
      (/
       2.0
       (*
        (/ (sin k) l)
        (*
         t_m
         (+
          (/ (* 2.0 (* t_m (* t_m (sin k)))) t_2)
          (* (* k k) (/ (sin k) t_2))))))
      (/
       (/
        (/ 2.0 t_m)
        (* (sin k) (* (+ 2.0 (/ k (* t_m (/ t_m k)))) (* (tan k) (/ t_m l)))))
       (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = l * cos(k);
	double tmp;
	if (t_m <= 9.2e+86) {
		tmp = 2.0 / ((sin(k) / l) * (t_m * (((2.0 * (t_m * (t_m * sin(k)))) / t_2) + ((k * k) * (sin(k) / t_2)))));
	} else {
		tmp = ((2.0 / t_m) / (sin(k) * ((2.0 + (k / (t_m * (t_m / k)))) * (tan(k) * (t_m / l))))) / (t_m / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = l * cos(k)
    if (t_m <= 9.2d+86) then
        tmp = 2.0d0 / ((sin(k) / l) * (t_m * (((2.0d0 * (t_m * (t_m * sin(k)))) / t_2) + ((k * k) * (sin(k) / t_2)))))
    else
        tmp = ((2.0d0 / t_m) / (sin(k) * ((2.0d0 + (k / (t_m * (t_m / k)))) * (tan(k) * (t_m / l))))) / (t_m / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = l * Math.cos(k);
	double tmp;
	if (t_m <= 9.2e+86) {
		tmp = 2.0 / ((Math.sin(k) / l) * (t_m * (((2.0 * (t_m * (t_m * Math.sin(k)))) / t_2) + ((k * k) * (Math.sin(k) / t_2)))));
	} else {
		tmp = ((2.0 / t_m) / (Math.sin(k) * ((2.0 + (k / (t_m * (t_m / k)))) * (Math.tan(k) * (t_m / l))))) / (t_m / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = l * math.cos(k)
	tmp = 0
	if t_m <= 9.2e+86:
		tmp = 2.0 / ((math.sin(k) / l) * (t_m * (((2.0 * (t_m * (t_m * math.sin(k)))) / t_2) + ((k * k) * (math.sin(k) / t_2)))))
	else:
		tmp = ((2.0 / t_m) / (math.sin(k) * ((2.0 + (k / (t_m * (t_m / k)))) * (math.tan(k) * (t_m / l))))) / (t_m / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(l * cos(k))
	tmp = 0.0
	if (t_m <= 9.2e+86)
		tmp = Float64(2.0 / Float64(Float64(sin(k) / l) * Float64(t_m * Float64(Float64(Float64(2.0 * Float64(t_m * Float64(t_m * sin(k)))) / t_2) + Float64(Float64(k * k) * Float64(sin(k) / t_2))))));
	else
		tmp = Float64(Float64(Float64(2.0 / t_m) / Float64(sin(k) * Float64(Float64(2.0 + Float64(k / Float64(t_m * Float64(t_m / k)))) * Float64(tan(k) * Float64(t_m / l))))) / Float64(t_m / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = l * cos(k);
	tmp = 0.0;
	if (t_m <= 9.2e+86)
		tmp = 2.0 / ((sin(k) / l) * (t_m * (((2.0 * (t_m * (t_m * sin(k)))) / t_2) + ((k * k) * (sin(k) / t_2)))));
	else
		tmp = ((2.0 / t_m) / (sin(k) * ((2.0 + (k / (t_m * (t_m / k)))) * (tan(k) * (t_m / l))))) / (t_m / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9.2e+86], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m * N[(N[(N[(2.0 * N[(t$95$m * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(N[(k * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(N[(2.0 + N[(k / N[(t$95$m * N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.19999999999999958e86

   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. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 51.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{\sin k}{\ell} \cdot \frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right) \cdot \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k}{\ell} \cdot \color{blue}{\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{\sin k}{\ell}\right), \color{blue}{\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin k, \ell\right), \left(\color{blue}{\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \left(\frac{\color{blue}{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right), \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right) \]

   6. Applied egg-rr 66.4%

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

   7. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \color{blue}{\left(t \cdot \left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k} + \frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)}\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(t, \color{blue}{\left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k} + \frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)}\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k}\right), \color{blue}{\left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)}\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left(\frac{2 \cdot \left({t}^{2} \cdot \sin k\right)}{\ell \cdot \cos k}\right), \left(\frac{\color{blue}{{k}^{2} \cdot \sin k}}{\ell \cdot \cos k}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left({t}^{2} \cdot \sin k\right)\right), \left(\ell \cdot \cos k\right)\right), \left(\frac{\color{blue}{{k}^{2} \cdot \sin k}}{\ell \cdot \cos k}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({t}^{2} \cdot \sin k\right)\right), \left(\ell \cdot \cos k\right)\right), \left(\frac{\color{blue}{{k}^{2}} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\left(t \cdot t\right) \cdot \sin k\right)\right), \left(\ell \cdot \cos k\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(t \cdot \left(t \cdot \sin k\right)\right)\right), \left(\ell \cdot \cos k\right)\right), \left(\frac{{k}^{\color{blue}{2}} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \left(t \cdot \sin k\right)\right)\right), \left(\ell \cdot \cos k\right)\right), \left(\frac{{k}^{\color{blue}{2}} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \sin k\right)\right)\right), \left(\ell \cdot \cos k\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right)\right)\right) \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right)\right)\right), \left(\ell \cdot \cos k\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \cos k\right)\right), \left(\frac{{k}^{2} \cdot \color{blue}{\sin k}}{\ell \cdot \cos k}\right)\right)\right)\right)\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \left(\frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k}\right)\right)\right)\right)\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \left({k}^{2} \cdot \color{blue}{\frac{\sin k}{\ell \cdot \cos k}}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \cos k}\right)}\right)\right)\right)\right)\right) \]

   9. Simplified 84.4%

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

   if 9.19999999999999958e86 < t

   1. Initial program 60.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 54.7%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{t}\right), \color{blue}{\left(\left(t \cdot t\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   6. Applied egg-rr 61.9%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{t \cdot \left(t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right)}{\color{blue}{\ell} \cdot \ell}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{t}{\ell} \cdot \color{blue}{\frac{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \color{blue}{\left(\frac{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}\right)}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\color{blue}{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}}{\ell}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right), \color{blue}{\ell}\right)\right)\right) \]

   8. Applied egg-rr 78.6%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right) \cdot t\right), \ell\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot t\right), \ell\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot t\right)\right), \ell\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \sin k\right), \left(\tan k \cdot t\right)\right), \ell\right)\right)\right) \]

   10. Applied egg-rr 89.3%

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

       1. *-commutative N/A

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

       2. associate-/r* N/A

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

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{t}}{\frac{\left(\sin k \cdot \left(2 + \frac{\frac{k}{t} \cdot k}{t}\right)\right) \cdot \left(\tan k \cdot t\right)}{\ell}}\right), \color{blue}{\left(\frac{t}{\ell}\right)}\right) \]

   12. Applied egg-rr 92.8%

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

Alternative 2: 87.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \frac{t\_m \cdot \sin k}{\ell \cdot \cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m \cdot \frac{t\_m}{\ell}}}{\sin k \cdot \left(2 + \frac{k}{t\_m \cdot \frac{t\_m}{k}}\right)}}{\tan k \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.8e-85)
    (/ 2.0 (* (/ (sin k) l) (* (* k k) (/ (* t_m (sin k)) (* l (cos k))))))
    (/
     (/ (/ 2.0 (* t_m (/ t_m l))) (* (sin k) (+ 2.0 (/ k (* t_m (/ t_m k))))))
     (* (tan k) (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.8e-85) {
		tmp = 2.0 / ((sin(k) / l) * ((k * k) * ((t_m * sin(k)) / (l * cos(k)))));
	} else {
		tmp = ((2.0 / (t_m * (t_m / l))) / (sin(k) * (2.0 + (k / (t_m * (t_m / k)))))) / (tan(k) * (t_m / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4.8d-85) then
        tmp = 2.0d0 / ((sin(k) / l) * ((k * k) * ((t_m * sin(k)) / (l * cos(k)))))
    else
        tmp = ((2.0d0 / (t_m * (t_m / l))) / (sin(k) * (2.0d0 + (k / (t_m * (t_m / k)))))) / (tan(k) * (t_m / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.8e-85) {
		tmp = 2.0 / ((Math.sin(k) / l) * ((k * k) * ((t_m * Math.sin(k)) / (l * Math.cos(k)))));
	} else {
		tmp = ((2.0 / (t_m * (t_m / l))) / (Math.sin(k) * (2.0 + (k / (t_m * (t_m / k)))))) / (Math.tan(k) * (t_m / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4.8e-85:
		tmp = 2.0 / ((math.sin(k) / l) * ((k * k) * ((t_m * math.sin(k)) / (l * math.cos(k)))))
	else:
		tmp = ((2.0 / (t_m * (t_m / l))) / (math.sin(k) * (2.0 + (k / (t_m * (t_m / k)))))) / (math.tan(k) * (t_m / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.8e-85)
		tmp = Float64(2.0 / Float64(Float64(sin(k) / l) * Float64(Float64(k * k) * Float64(Float64(t_m * sin(k)) / Float64(l * cos(k))))));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(t_m * Float64(t_m / l))) / Float64(sin(k) * Float64(2.0 + Float64(k / Float64(t_m * Float64(t_m / k)))))) / Float64(tan(k) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4.8e-85)
		tmp = 2.0 / ((sin(k) / l) * ((k * k) * ((t_m * sin(k)) / (l * cos(k)))));
	else
		tmp = ((2.0 / (t_m * (t_m / l))) / (sin(k) * (2.0 + (k / (t_m * (t_m / k)))))) / (tan(k) * (t_m / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.8e-85], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(2.0 + N[(k / N[(t$95$m * N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.8000000000000001e-85

   1. Initial program 52.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 47.5%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{\sin k}{\ell} \cdot \frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right) \cdot \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k}{\ell} \cdot \color{blue}{\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{\sin k}{\ell}\right), \color{blue}{\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin k, \ell\right), \left(\color{blue}{\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \left(\frac{\color{blue}{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right), \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right) \]

   6. Applied egg-rr 63.7%

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

   7. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{\ell \cdot \cos k}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \left({k}^{2} \cdot \color{blue}{\frac{t \cdot \sin k}{\ell \cdot \cos k}}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{\left(\frac{t \cdot \sin k}{\ell \cdot \cos k}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\left(k \cdot k\right), \left(\frac{\color{blue}{t \cdot \sin k}}{\ell \cdot \cos k}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{\color{blue}{t \cdot \sin k}}{\ell \cdot \cos k}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\left(t \cdot \sin k\right), \color{blue}{\left(\ell \cdot \cos k\right)}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \sin k\right), \left(\color{blue}{\ell} \cdot \cos k\right)\right)\right)\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \left(\ell \cdot \cos k\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\cos k}\right)\right)\right)\right)\right) \]

      9. cos-lowering-cos.f64 68.7%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right)\right)\right) \]

   9. Simplified 68.7%

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

   if 4.8000000000000001e-85 < t

   1. Initial program 66.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 61.9%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{t}\right), \color{blue}{\left(\left(t \cdot t\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   6. Applied egg-rr 65.8%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{t \cdot \left(t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right)}{\color{blue}{\ell} \cdot \ell}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{t}{\ell} \cdot \color{blue}{\frac{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \color{blue}{\left(\frac{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}\right)}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\color{blue}{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}}{\ell}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right), \color{blue}{\ell}\right)\right)\right) \]

   8. Applied egg-rr 79.8%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right) \cdot t\right), \ell\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot t\right), \ell\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot t\right)\right), \ell\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \sin k\right), \left(\tan k \cdot t\right)\right), \ell\right)\right)\right) \]

   10. Applied egg-rr 87.3%

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

       1. associate-/r* N/A

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

       2. associate-/l* N/A

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

       3. associate-/r* N/A

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

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{2}{t}}{\frac{t}{\ell}}}{\sin k \cdot \left(2 + \frac{\frac{k}{t} \cdot k}{t}\right)}\right), \color{blue}{\left(\frac{\tan k \cdot t}{\ell}\right)}\right) \]

   12. Applied egg-rr 91.3%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \frac{t \cdot \sin k}{\ell \cdot \cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{t \cdot \frac{t}{\ell}}}{\sin k \cdot \left(2 + \frac{k}{t \cdot \frac{t}{k}}\right)}}{\tan k \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.85 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m}}{\frac{k \cdot k}{\ell}}}{\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t\_m \cdot t\_m\right) \cdot 0.3333333333333333\right) + 2 \cdot \left(t\_m \cdot t\_m\right)}{\ell}}\\ \mathbf{elif}\;t\_m \leq 1.9 \cdot 10^{+92}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(\left(\left(2 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right) \cdot \frac{\tan k}{\ell}\right) \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m}}{\sin k \cdot \left(\left(2 + \frac{k}{t\_m \cdot \frac{t\_m}{k}}\right) \cdot \left(\tan k \cdot \frac{t\_m}{\ell}\right)\right)}}{\frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.85e-115)
    (/
     (/ (/ 2.0 t_m) (/ (* k k) l))
     (/
      (+
       (* (* k k) (+ 1.0 (* (* t_m t_m) 0.3333333333333333)))
       (* 2.0 (* t_m t_m)))
      l))
    (if (<= t_m 1.9e+92)
      (/
       2.0
       (*
        (/ (sin k) l)
        (*
         (* (+ 2.0 (/ (/ k t_m) (/ t_m k))) (/ (tan k) l))
         (* t_m (* t_m t_m)))))
      (/
       (/
        (/ 2.0 t_m)
        (* (sin k) (* (+ 2.0 (/ k (* t_m (/ t_m k)))) (* (tan k) (/ t_m l)))))
       (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.85e-115) {
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l);
	} else if (t_m <= 1.9e+92) {
		tmp = 2.0 / ((sin(k) / l) * (((2.0 + ((k / t_m) / (t_m / k))) * (tan(k) / l)) * (t_m * (t_m * t_m))));
	} else {
		tmp = ((2.0 / t_m) / (sin(k) * ((2.0 + (k / (t_m * (t_m / k)))) * (tan(k) * (t_m / l))))) / (t_m / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.85d-115) then
        tmp = ((2.0d0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0d0 + ((t_m * t_m) * 0.3333333333333333d0))) + (2.0d0 * (t_m * t_m))) / l)
    else if (t_m <= 1.9d+92) then
        tmp = 2.0d0 / ((sin(k) / l) * (((2.0d0 + ((k / t_m) / (t_m / k))) * (tan(k) / l)) * (t_m * (t_m * t_m))))
    else
        tmp = ((2.0d0 / t_m) / (sin(k) * ((2.0d0 + (k / (t_m * (t_m / k)))) * (tan(k) * (t_m / l))))) / (t_m / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.85e-115) {
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l);
	} else if (t_m <= 1.9e+92) {
		tmp = 2.0 / ((Math.sin(k) / l) * (((2.0 + ((k / t_m) / (t_m / k))) * (Math.tan(k) / l)) * (t_m * (t_m * t_m))));
	} else {
		tmp = ((2.0 / t_m) / (Math.sin(k) * ((2.0 + (k / (t_m * (t_m / k)))) * (Math.tan(k) * (t_m / l))))) / (t_m / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.85e-115:
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l)
	elif t_m <= 1.9e+92:
		tmp = 2.0 / ((math.sin(k) / l) * (((2.0 + ((k / t_m) / (t_m / k))) * (math.tan(k) / l)) * (t_m * (t_m * t_m))))
	else:
		tmp = ((2.0 / t_m) / (math.sin(k) * ((2.0 + (k / (t_m * (t_m / k)))) * (math.tan(k) * (t_m / l))))) / (t_m / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.85e-115)
		tmp = Float64(Float64(Float64(2.0 / t_m) / Float64(Float64(k * k) / l)) / Float64(Float64(Float64(Float64(k * k) * Float64(1.0 + Float64(Float64(t_m * t_m) * 0.3333333333333333))) + Float64(2.0 * Float64(t_m * t_m))) / l));
	elseif (t_m <= 1.9e+92)
		tmp = Float64(2.0 / Float64(Float64(sin(k) / l) * Float64(Float64(Float64(2.0 + Float64(Float64(k / t_m) / Float64(t_m / k))) * Float64(tan(k) / l)) * Float64(t_m * Float64(t_m * t_m)))));
	else
		tmp = Float64(Float64(Float64(2.0 / t_m) / Float64(sin(k) * Float64(Float64(2.0 + Float64(k / Float64(t_m * Float64(t_m / k)))) * Float64(tan(k) * Float64(t_m / l))))) / Float64(t_m / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.85e-115)
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l);
	elseif (t_m <= 1.9e+92)
		tmp = 2.0 / ((sin(k) / l) * (((2.0 + ((k / t_m) / (t_m / k))) * (tan(k) / l)) * (t_m * (t_m * t_m))));
	else
		tmp = ((2.0 / t_m) / (sin(k) * ((2.0 + (k / (t_m * (t_m / k)))) * (tan(k) * (t_m / l))))) / (t_m / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.85e-115], N[(N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(k * k), $MachinePrecision] * N[(1.0 + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.9e+92], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] / N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(N[(2.0 + N[(k / N[(t$95$m * N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.85e-115

   1. Initial program 51.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 46.5%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{t}\right), \color{blue}{\left(\left(t \cdot t\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   6. Applied egg-rr 50.6%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\color{blue}{\left({k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\ell}, \ell\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right) + 2 \cdot {t}^{2}\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left({k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({t}^{2} \cdot \left(\frac{1}{{t}^{2}} + \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({t}^{2} \cdot \frac{1}{{t}^{2}} + {t}^{2} \cdot \frac{1}{3}\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      11. rgt-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(1 + {t}^{2} \cdot \frac{1}{3}\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \left({t}^{2} \cdot \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({t}^{2}\right), \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(t \cdot t\right), \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \mathsf{*.f64}\left(2, \left({t}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \mathsf{*.f64}\left(2, \left(t \cdot t\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      18. *-lowering-*.f64 58.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

   9. Simplified 58.0%

      \[\leadsto \frac{\frac{2}{t}}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot 0.3333333333333333\right) + 2 \cdot \left(t \cdot t\right)\right)}}{\ell \cdot \ell}} \]
   10. Step-by-step derivation

       1. times-frac N/A

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

       2. associate-/r* N/A

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

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{t}}{\frac{k \cdot k}{\ell}}\right), \color{blue}{\left(\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + 2 \cdot \left(t \cdot t\right)}{\ell}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{t}\right), \left(\frac{k \cdot k}{\ell}\right)\right), \left(\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + 2 \cdot \left(t \cdot t\right)}}{\ell}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{k \cdot k}{\ell}\right)\right), \left(\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right)} + 2 \cdot \left(t \cdot t\right)}{\ell}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(k \cdot k\right), \ell\right)\right), \left(\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + \color{blue}{2 \cdot \left(t \cdot t\right)}}{\ell}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right)\right), \left(\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + \color{blue}{2} \cdot \left(t \cdot t\right)}{\ell}\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right)\right), \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + 2 \cdot \left(t \cdot t\right)\right), \color{blue}{\ell}\right)\right) \]

   11. Applied egg-rr 67.3%

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

   if 1.85e-115 < t < 1.9e92

   1. Initial program 71.5%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 76.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{\sin k}{\ell} \cdot \frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right) \cdot \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k}{\ell} \cdot \color{blue}{\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{\sin k}{\ell}\right), \color{blue}{\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin k, \ell\right), \left(\color{blue}{\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \left(\frac{\color{blue}{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right), \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right) \]

   6. Applied egg-rr 85.3%

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

   if 1.9e92 < t

   1. Initial program 61.3%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 53.1%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{t}\right), \color{blue}{\left(\left(t \cdot t\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   6. Applied egg-rr 59.0%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{t \cdot \left(t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right)}{\color{blue}{\ell} \cdot \ell}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{t}{\ell} \cdot \color{blue}{\frac{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \color{blue}{\left(\frac{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}\right)}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\color{blue}{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}}{\ell}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right), \color{blue}{\ell}\right)\right)\right) \]

   8. Applied egg-rr 77.0%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right) \cdot t\right), \ell\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot t\right), \ell\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot t\right)\right), \ell\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \sin k\right), \left(\tan k \cdot t\right)\right), \ell\right)\right)\right) \]

   10. Applied egg-rr 88.5%

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

       1. *-commutative N/A

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

       2. associate-/r* N/A

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

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{t}}{\frac{\left(\sin k \cdot \left(2 + \frac{\frac{k}{t} \cdot k}{t}\right)\right) \cdot \left(\tan k \cdot t\right)}{\ell}}\right), \color{blue}{\left(\frac{t}{\ell}\right)}\right) \]

   12. Applied egg-rr 92.3%

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

Alternative 4: 82.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.85 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m}}{\frac{k \cdot k}{\ell}}}{\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t\_m \cdot t\_m\right) \cdot 0.3333333333333333\right) + 2 \cdot \left(t\_m \cdot t\_m\right)}{\ell}}\\ \mathbf{elif}\;t\_m \leq 3.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(\left(\left(2 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right) \cdot \frac{\tan k}{\ell}\right) \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{2}{t\_m}}{\left(\sin k \cdot \frac{t\_m}{\ell}\right) \cdot \left(\left(2 + \frac{k}{t\_m \cdot \frac{t\_m}{k}}\right) \cdot \left(t\_m \cdot \tan k\right)\right)}\\ \end{array} \end{array} \]

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 (<= t_m 1.85e-115)
    (/
     (/ (/ 2.0 t_m) (/ (* k k) l))
     (/
      (+
       (* (* k k) (+ 1.0 (* (* t_m t_m) 0.3333333333333333)))
       (* 2.0 (* t_m t_m)))
      l))
    (if (<= t_m 3.6e+92)
      (/
       2.0
       (*
        (/ (sin k) l)
        (*
         (* (+ 2.0 (/ (/ k t_m) (/ t_m k))) (/ (tan k) l))
         (* t_m (* t_m t_m)))))
      (*
       l
       (/
        (/ 2.0 t_m)
        (*
         (* (sin k) (/ t_m l))
         (* (+ 2.0 (/ k (* t_m (/ t_m k)))) (* t_m (tan k))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.85e-115) {
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l);
	} else if (t_m <= 3.6e+92) {
		tmp = 2.0 / ((sin(k) / l) * (((2.0 + ((k / t_m) / (t_m / k))) * (tan(k) / l)) * (t_m * (t_m * t_m))));
	} else {
		tmp = l * ((2.0 / t_m) / ((sin(k) * (t_m / l)) * ((2.0 + (k / (t_m * (t_m / k)))) * (t_m * tan(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 <= 1.85d-115) then
        tmp = ((2.0d0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0d0 + ((t_m * t_m) * 0.3333333333333333d0))) + (2.0d0 * (t_m * t_m))) / l)
    else if (t_m <= 3.6d+92) then
        tmp = 2.0d0 / ((sin(k) / l) * (((2.0d0 + ((k / t_m) / (t_m / k))) * (tan(k) / l)) * (t_m * (t_m * t_m))))
    else
        tmp = l * ((2.0d0 / t_m) / ((sin(k) * (t_m / l)) * ((2.0d0 + (k / (t_m * (t_m / k)))) * (t_m * tan(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 <= 1.85e-115) {
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l);
	} else if (t_m <= 3.6e+92) {
		tmp = 2.0 / ((Math.sin(k) / l) * (((2.0 + ((k / t_m) / (t_m / k))) * (Math.tan(k) / l)) * (t_m * (t_m * t_m))));
	} else {
		tmp = l * ((2.0 / t_m) / ((Math.sin(k) * (t_m / l)) * ((2.0 + (k / (t_m * (t_m / k)))) * (t_m * Math.tan(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 <= 1.85e-115:
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l)
	elif t_m <= 3.6e+92:
		tmp = 2.0 / ((math.sin(k) / l) * (((2.0 + ((k / t_m) / (t_m / k))) * (math.tan(k) / l)) * (t_m * (t_m * t_m))))
	else:
		tmp = l * ((2.0 / t_m) / ((math.sin(k) * (t_m / l)) * ((2.0 + (k / (t_m * (t_m / k)))) * (t_m * math.tan(k)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.85e-115)
		tmp = Float64(Float64(Float64(2.0 / t_m) / Float64(Float64(k * k) / l)) / Float64(Float64(Float64(Float64(k * k) * Float64(1.0 + Float64(Float64(t_m * t_m) * 0.3333333333333333))) + Float64(2.0 * Float64(t_m * t_m))) / l));
	elseif (t_m <= 3.6e+92)
		tmp = Float64(2.0 / Float64(Float64(sin(k) / l) * Float64(Float64(Float64(2.0 + Float64(Float64(k / t_m) / Float64(t_m / k))) * Float64(tan(k) / l)) * Float64(t_m * Float64(t_m * t_m)))));
	else
		tmp = Float64(l * Float64(Float64(2.0 / t_m) / Float64(Float64(sin(k) * Float64(t_m / l)) * Float64(Float64(2.0 + Float64(k / Float64(t_m * Float64(t_m / k)))) * Float64(t_m * tan(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 <= 1.85e-115)
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l);
	elseif (t_m <= 3.6e+92)
		tmp = 2.0 / ((sin(k) / l) * (((2.0 + ((k / t_m) / (t_m / k))) * (tan(k) / l)) * (t_m * (t_m * t_m))));
	else
		tmp = l * ((2.0 / t_m) / ((sin(k) * (t_m / l)) * ((2.0 + (k / (t_m * (t_m / k)))) * (t_m * tan(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, 1.85e-115], N[(N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(k * k), $MachinePrecision] * N[(1.0 + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.6e+92], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] / N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + N[(k / N[(t$95$m * N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.85e-115

   1. Initial program 51.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 46.5%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{t}\right), \color{blue}{\left(\left(t \cdot t\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   6. Applied egg-rr 50.6%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\color{blue}{\left({k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\ell}, \ell\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right) + 2 \cdot {t}^{2}\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left({k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({t}^{2} \cdot \left(\frac{1}{{t}^{2}} + \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({t}^{2} \cdot \frac{1}{{t}^{2}} + {t}^{2} \cdot \frac{1}{3}\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      11. rgt-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(1 + {t}^{2} \cdot \frac{1}{3}\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \left({t}^{2} \cdot \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({t}^{2}\right), \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(t \cdot t\right), \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \mathsf{*.f64}\left(2, \left({t}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \mathsf{*.f64}\left(2, \left(t \cdot t\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      18. *-lowering-*.f64 58.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

   9. Simplified 58.0%

      \[\leadsto \frac{\frac{2}{t}}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot 0.3333333333333333\right) + 2 \cdot \left(t \cdot t\right)\right)}}{\ell \cdot \ell}} \]
   10. Step-by-step derivation

       1. times-frac N/A

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

       2. associate-/r* N/A

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

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{t}}{\frac{k \cdot k}{\ell}}\right), \color{blue}{\left(\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + 2 \cdot \left(t \cdot t\right)}{\ell}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{t}\right), \left(\frac{k \cdot k}{\ell}\right)\right), \left(\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + 2 \cdot \left(t \cdot t\right)}}{\ell}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{k \cdot k}{\ell}\right)\right), \left(\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right)} + 2 \cdot \left(t \cdot t\right)}{\ell}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(k \cdot k\right), \ell\right)\right), \left(\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + \color{blue}{2 \cdot \left(t \cdot t\right)}}{\ell}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right)\right), \left(\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + \color{blue}{2} \cdot \left(t \cdot t\right)}{\ell}\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right)\right), \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + 2 \cdot \left(t \cdot t\right)\right), \color{blue}{\ell}\right)\right) \]

   11. Applied egg-rr 67.3%

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

   if 1.85e-115 < t < 3.6e92

   1. Initial program 72.3%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 74.8%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{\sin k}{\ell} \cdot \frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right) \cdot \left(\color{blue}{t} \cdot \left(t \cdot t\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\sin k}{\ell} \cdot \color{blue}{\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{\sin k}{\ell}\right), \color{blue}{\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin k, \ell\right), \left(\color{blue}{\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \left(\frac{\color{blue}{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}}{\ell} \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(k\right), \ell\right), \mathsf{*.f64}\left(\left(\frac{\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\ell}\right), \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right) \]

   6. Applied egg-rr 85.7%

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

   if 3.6e92 < t

   1. Initial program 60.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 54.1%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{t}\right), \color{blue}{\left(\left(t \cdot t\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   6. Applied egg-rr 58.2%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{t \cdot \left(t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right)}{\color{blue}{\ell} \cdot \ell}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{t}{\ell} \cdot \color{blue}{\frac{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \color{blue}{\left(\frac{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}\right)}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\color{blue}{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}}{\ell}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right), \color{blue}{\ell}\right)\right)\right) \]

   8. Applied egg-rr 76.5%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right) \cdot t\right), \ell\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot t\right), \ell\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot t\right)\right), \ell\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \sin k\right), \left(\tan k \cdot t\right)\right), \ell\right)\right)\right) \]

   10. Applied egg-rr 88.3%

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

       1. associate-*r/ N/A

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

       2. associate-/r/ N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{t}}{\frac{t}{\ell} \cdot \left(\left(\sin k \cdot \left(2 + \frac{\frac{k}{t} \cdot k}{t}\right)\right) \cdot \left(\tan k \cdot t\right)\right)}\right), \color{blue}{\ell}\right) \]

   12. Applied egg-rr 90.5%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.85 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{\frac{2}{t}}{\frac{k \cdot k}{\ell}}}{\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot 0.3333333333333333\right) + 2 \cdot \left(t \cdot t\right)}{\ell}}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(\left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \frac{\tan k}{\ell}\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{2}{t}}{\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \left(\left(2 + \frac{k}{t \cdot \frac{t}{k}}\right) \cdot \left(t \cdot \tan k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.9 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m}}{\frac{k \cdot k}{\ell}}}{\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t\_m \cdot t\_m\right) \cdot 0.3333333333333333\right) + 2 \cdot \left(t\_m \cdot t\_m\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m \cdot \frac{t\_m}{\ell}}}{\sin k \cdot \left(2 + \frac{k}{t\_m \cdot \frac{t\_m}{k}}\right)}}{\tan k \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.9e-116)
    (/
     (/ (/ 2.0 t_m) (/ (* k k) l))
     (/
      (+
       (* (* k k) (+ 1.0 (* (* t_m t_m) 0.3333333333333333)))
       (* 2.0 (* t_m t_m)))
      l))
    (/
     (/ (/ 2.0 (* t_m (/ t_m l))) (* (sin k) (+ 2.0 (/ k (* t_m (/ t_m k))))))
     (* (tan k) (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.9e-116) {
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l);
	} else {
		tmp = ((2.0 / (t_m * (t_m / l))) / (sin(k) * (2.0 + (k / (t_m * (t_m / k)))))) / (tan(k) * (t_m / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4.9d-116) then
        tmp = ((2.0d0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0d0 + ((t_m * t_m) * 0.3333333333333333d0))) + (2.0d0 * (t_m * t_m))) / l)
    else
        tmp = ((2.0d0 / (t_m * (t_m / l))) / (sin(k) * (2.0d0 + (k / (t_m * (t_m / k)))))) / (tan(k) * (t_m / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.9e-116) {
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l);
	} else {
		tmp = ((2.0 / (t_m * (t_m / l))) / (Math.sin(k) * (2.0 + (k / (t_m * (t_m / k)))))) / (Math.tan(k) * (t_m / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4.9e-116:
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l)
	else:
		tmp = ((2.0 / (t_m * (t_m / l))) / (math.sin(k) * (2.0 + (k / (t_m * (t_m / k)))))) / (math.tan(k) * (t_m / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.9e-116)
		tmp = Float64(Float64(Float64(2.0 / t_m) / Float64(Float64(k * k) / l)) / Float64(Float64(Float64(Float64(k * k) * Float64(1.0 + Float64(Float64(t_m * t_m) * 0.3333333333333333))) + Float64(2.0 * Float64(t_m * t_m))) / l));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(t_m * Float64(t_m / l))) / Float64(sin(k) * Float64(2.0 + Float64(k / Float64(t_m * Float64(t_m / k)))))) / Float64(tan(k) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4.9e-116)
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l);
	else
		tmp = ((2.0 / (t_m * (t_m / l))) / (sin(k) * (2.0 + (k / (t_m * (t_m / k)))))) / (tan(k) * (t_m / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.9e-116], N[(N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(k * k), $MachinePrecision] * N[(1.0 + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(2.0 + N[(k / N[(t$95$m * N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.89999999999999977e-116

   1. Initial program 51.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 46.5%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{t}\right), \color{blue}{\left(\left(t \cdot t\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   6. Applied egg-rr 50.6%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\color{blue}{\left({k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\ell}, \ell\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right) + 2 \cdot {t}^{2}\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left({k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({t}^{2} \cdot \left(\frac{1}{{t}^{2}} + \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({t}^{2} \cdot \frac{1}{{t}^{2}} + {t}^{2} \cdot \frac{1}{3}\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      11. rgt-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(1 + {t}^{2} \cdot \frac{1}{3}\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \left({t}^{2} \cdot \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({t}^{2}\right), \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(t \cdot t\right), \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \mathsf{*.f64}\left(2, \left({t}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \mathsf{*.f64}\left(2, \left(t \cdot t\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      18. *-lowering-*.f64 58.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

   9. Simplified 58.0%

      \[\leadsto \frac{\frac{2}{t}}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot 0.3333333333333333\right) + 2 \cdot \left(t \cdot t\right)\right)}}{\ell \cdot \ell}} \]
   10. Step-by-step derivation

       1. times-frac N/A

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

       2. associate-/r* N/A

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

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{t}}{\frac{k \cdot k}{\ell}}\right), \color{blue}{\left(\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + 2 \cdot \left(t \cdot t\right)}{\ell}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{t}\right), \left(\frac{k \cdot k}{\ell}\right)\right), \left(\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + 2 \cdot \left(t \cdot t\right)}}{\ell}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{k \cdot k}{\ell}\right)\right), \left(\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right)} + 2 \cdot \left(t \cdot t\right)}{\ell}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(k \cdot k\right), \ell\right)\right), \left(\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + \color{blue}{2 \cdot \left(t \cdot t\right)}}{\ell}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right)\right), \left(\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + \color{blue}{2} \cdot \left(t \cdot t\right)}{\ell}\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right)\right), \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + 2 \cdot \left(t \cdot t\right)\right), \color{blue}{\ell}\right)\right) \]

   11. Applied egg-rr 67.3%

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

   if 4.89999999999999977e-116 < t

   1. Initial program 65.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 62.6%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{t}\right), \color{blue}{\left(\left(t \cdot t\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   6. Applied egg-rr 66.2%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{t \cdot \left(t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right)}{\color{blue}{\ell} \cdot \ell}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{t}{\ell} \cdot \color{blue}{\frac{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \color{blue}{\left(\frac{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}\right)}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\color{blue}{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}}{\ell}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right), \color{blue}{\ell}\right)\right)\right) \]

   8. Applied egg-rr 79.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right) \cdot t\right), \ell\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot t\right), \ell\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot t\right)\right), \ell\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \sin k\right), \left(\tan k \cdot t\right)\right), \ell\right)\right)\right) \]

   10. Applied egg-rr 86.0%

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

       1. associate-/r* N/A

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

       2. associate-/l* N/A

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

       3. associate-/r* N/A

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

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{2}{t}}{\frac{t}{\ell}}}{\sin k \cdot \left(2 + \frac{\frac{k}{t} \cdot k}{t}\right)}\right), \color{blue}{\left(\frac{\tan k \cdot t}{\ell}\right)}\right) \]

   12. Applied egg-rr 89.6%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.9 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{\frac{2}{t}}{\frac{k \cdot k}{\ell}}}{\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot 0.3333333333333333\right) + 2 \cdot \left(t \cdot t\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{t \cdot \frac{t}{\ell}}}{\sin k \cdot \left(2 + \frac{k}{t \cdot \frac{t}{k}}\right)}}{\tan k \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.8 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m}}{\frac{k \cdot k}{\ell}}}{\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t\_m \cdot t\_m\right) \cdot 0.3333333333333333\right) + 2 \cdot \left(t\_m \cdot t\_m\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{2}{t\_m}}{\left(\sin k \cdot \frac{t\_m}{\ell}\right) \cdot \left(\left(2 + \frac{k}{t\_m \cdot \frac{t\_m}{k}}\right) \cdot \left(t\_m \cdot \tan k\right)\right)}\\ \end{array} \end{array} \]

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 (<= t_m 5.8e-57)
    (/
     (/ (/ 2.0 t_m) (/ (* k k) l))
     (/
      (+
       (* (* k k) (+ 1.0 (* (* t_m t_m) 0.3333333333333333)))
       (* 2.0 (* t_m t_m)))
      l))
    (*
     l
     (/
      (/ 2.0 t_m)
      (*
       (* (sin k) (/ t_m l))
       (* (+ 2.0 (/ k (* t_m (/ t_m k)))) (* t_m (tan k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.8e-57) {
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l);
	} else {
		tmp = l * ((2.0 / t_m) / ((sin(k) * (t_m / l)) * ((2.0 + (k / (t_m * (t_m / k)))) * (t_m * tan(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 <= 5.8d-57) then
        tmp = ((2.0d0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0d0 + ((t_m * t_m) * 0.3333333333333333d0))) + (2.0d0 * (t_m * t_m))) / l)
    else
        tmp = l * ((2.0d0 / t_m) / ((sin(k) * (t_m / l)) * ((2.0d0 + (k / (t_m * (t_m / k)))) * (t_m * tan(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 <= 5.8e-57) {
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l);
	} else {
		tmp = l * ((2.0 / t_m) / ((Math.sin(k) * (t_m / l)) * ((2.0 + (k / (t_m * (t_m / k)))) * (t_m * Math.tan(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 <= 5.8e-57:
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l)
	else:
		tmp = l * ((2.0 / t_m) / ((math.sin(k) * (t_m / l)) * ((2.0 + (k / (t_m * (t_m / k)))) * (t_m * math.tan(k)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.8e-57)
		tmp = Float64(Float64(Float64(2.0 / t_m) / Float64(Float64(k * k) / l)) / Float64(Float64(Float64(Float64(k * k) * Float64(1.0 + Float64(Float64(t_m * t_m) * 0.3333333333333333))) + Float64(2.0 * Float64(t_m * t_m))) / l));
	else
		tmp = Float64(l * Float64(Float64(2.0 / t_m) / Float64(Float64(sin(k) * Float64(t_m / l)) * Float64(Float64(2.0 + Float64(k / Float64(t_m * Float64(t_m / k)))) * Float64(t_m * tan(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 <= 5.8e-57)
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l);
	else
		tmp = l * ((2.0 / t_m) / ((sin(k) * (t_m / l)) * ((2.0 + (k / (t_m * (t_m / k)))) * (t_m * tan(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, 5.8e-57], N[(N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(k * k), $MachinePrecision] * N[(1.0 + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + N[(k / N[(t$95$m * N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.8000000000000005e-57

   1. Initial program 52.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 47.5%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{t}\right), \color{blue}{\left(\left(t \cdot t\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   6. Applied egg-rr 51.4%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\color{blue}{\left({k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\ell}, \ell\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right) + 2 \cdot {t}^{2}\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left({k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({t}^{2} \cdot \left(\frac{1}{{t}^{2}} + \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({t}^{2} \cdot \frac{1}{{t}^{2}} + {t}^{2} \cdot \frac{1}{3}\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      11. rgt-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(1 + {t}^{2} \cdot \frac{1}{3}\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \left({t}^{2} \cdot \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({t}^{2}\right), \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(t \cdot t\right), \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \mathsf{*.f64}\left(2, \left({t}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \mathsf{*.f64}\left(2, \left(t \cdot t\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      18. *-lowering-*.f64 58.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

   9. Simplified 58.5%

      \[\leadsto \frac{\frac{2}{t}}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot 0.3333333333333333\right) + 2 \cdot \left(t \cdot t\right)\right)}}{\ell \cdot \ell}} \]
   10. Step-by-step derivation

       1. times-frac N/A

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

       2. associate-/r* N/A

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

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{t}}{\frac{k \cdot k}{\ell}}\right), \color{blue}{\left(\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + 2 \cdot \left(t \cdot t\right)}{\ell}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{t}\right), \left(\frac{k \cdot k}{\ell}\right)\right), \left(\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + 2 \cdot \left(t \cdot t\right)}}{\ell}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{k \cdot k}{\ell}\right)\right), \left(\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right)} + 2 \cdot \left(t \cdot t\right)}{\ell}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(k \cdot k\right), \ell\right)\right), \left(\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + \color{blue}{2 \cdot \left(t \cdot t\right)}}{\ell}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right)\right), \left(\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + \color{blue}{2} \cdot \left(t \cdot t\right)}{\ell}\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right)\right), \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + 2 \cdot \left(t \cdot t\right)\right), \color{blue}{\ell}\right)\right) \]

   11. Applied egg-rr 67.3%

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

   if 5.8000000000000005e-57 < t

   1. Initial program 66.5%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 62.1%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{t}\right), \color{blue}{\left(\left(t \cdot t\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   6. Applied egg-rr 66.1%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{t \cdot \left(t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right)}{\color{blue}{\ell} \cdot \ell}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{t}{\ell} \cdot \color{blue}{\frac{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \color{blue}{\left(\frac{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}\right)}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\color{blue}{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}}{\ell}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right), \color{blue}{\ell}\right)\right)\right) \]

   8. Applied egg-rr 80.5%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right) \cdot t\right), \ell\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot t\right), \ell\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot t\right)\right), \ell\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \sin k\right), \left(\tan k \cdot t\right)\right), \ell\right)\right)\right) \]

   10. Applied egg-rr 88.3%

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

       1. associate-*r/ N/A

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

       2. associate-/r/ N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{t}}{\frac{t}{\ell} \cdot \left(\left(\sin k \cdot \left(2 + \frac{\frac{k}{t} \cdot k}{t}\right)\right) \cdot \left(\tan k \cdot t\right)\right)}\right), \color{blue}{\ell}\right) \]

   12. Applied egg-rr 89.9%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{\frac{2}{t}}{\frac{k \cdot k}{\ell}}}{\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot 0.3333333333333333\right) + 2 \cdot \left(t \cdot t\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{2}{t}}{\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \left(\left(2 + \frac{k}{t \cdot \frac{t}{k}}\right) \cdot \left(t \cdot \tan k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-77}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m}}{\frac{k \cdot k}{\ell}}}{\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t\_m \cdot t\_m\right) \cdot 0.3333333333333333\right) + 2 \cdot \left(t\_m \cdot t\_m\right)}{\ell}}\\ \mathbf{elif}\;t\_m \leq 6.4 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot k}}{t\_m \cdot t\_m} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t\_m}}{\frac{t\_m}{\ell} \cdot \frac{\left(k \cdot \left(2 + \frac{k \cdot \frac{k}{t\_m}}{t\_m}\right)\right) \cdot \left(t\_m \cdot \tan k\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1e-77)
    (/
     (/ (/ 2.0 t_m) (/ (* k k) l))
     (/
      (+
       (* (* k k) (+ 1.0 (* (* t_m t_m) 0.3333333333333333)))
       (* 2.0 (* t_m t_m)))
      l))
    (if (<= t_m 6.4e+152)
      (* (/ (/ l (* t_m k)) (* t_m t_m)) (/ l k))
      (/
       (/ 2.0 t_m)
       (*
        (/ t_m l)
        (/ (* (* k (+ 2.0 (/ (* k (/ k t_m)) t_m))) (* t_m (tan 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 <= 1e-77) {
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l);
	} else if (t_m <= 6.4e+152) {
		tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / k);
	} else {
		tmp = (2.0 / t_m) / ((t_m / l) * (((k * (2.0 + ((k * (k / t_m)) / t_m))) * (t_m * tan(k))) / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1d-77) then
        tmp = ((2.0d0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0d0 + ((t_m * t_m) * 0.3333333333333333d0))) + (2.0d0 * (t_m * t_m))) / l)
    else if (t_m <= 6.4d+152) then
        tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / k)
    else
        tmp = (2.0d0 / t_m) / ((t_m / l) * (((k * (2.0d0 + ((k * (k / t_m)) / t_m))) * (t_m * tan(k))) / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1e-77) {
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l);
	} else if (t_m <= 6.4e+152) {
		tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / k);
	} else {
		tmp = (2.0 / t_m) / ((t_m / l) * (((k * (2.0 + ((k * (k / t_m)) / t_m))) * (t_m * Math.tan(k))) / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1e-77:
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l)
	elif t_m <= 6.4e+152:
		tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / k)
	else:
		tmp = (2.0 / t_m) / ((t_m / l) * (((k * (2.0 + ((k * (k / t_m)) / t_m))) * (t_m * math.tan(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 <= 1e-77)
		tmp = Float64(Float64(Float64(2.0 / t_m) / Float64(Float64(k * k) / l)) / Float64(Float64(Float64(Float64(k * k) * Float64(1.0 + Float64(Float64(t_m * t_m) * 0.3333333333333333))) + Float64(2.0 * Float64(t_m * t_m))) / l));
	elseif (t_m <= 6.4e+152)
		tmp = Float64(Float64(Float64(l / Float64(t_m * k)) / Float64(t_m * t_m)) * Float64(l / k));
	else
		tmp = Float64(Float64(2.0 / t_m) / Float64(Float64(t_m / l) * Float64(Float64(Float64(k * Float64(2.0 + Float64(Float64(k * Float64(k / t_m)) / t_m))) * Float64(t_m * tan(k))) / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1e-77)
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l);
	elseif (t_m <= 6.4e+152)
		tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / k);
	else
		tmp = (2.0 / t_m) / ((t_m / l) * (((k * (2.0 + ((k * (k / t_m)) / t_m))) * (t_m * tan(k))) / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1e-77], N[(N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(k * k), $MachinePrecision] * N[(1.0 + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.4e+152], N[(N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(k * N[(2.0 + N[(N[(k * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 9.9999999999999993e-78

   1. Initial program 52.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 47.5%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{t}\right), \color{blue}{\left(\left(t \cdot t\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   6. Applied egg-rr 51.4%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\color{blue}{\left({k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\ell}, \ell\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right) + 2 \cdot {t}^{2}\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left({k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({t}^{2} \cdot \left(\frac{1}{{t}^{2}} + \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({t}^{2} \cdot \frac{1}{{t}^{2}} + {t}^{2} \cdot \frac{1}{3}\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      11. rgt-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(1 + {t}^{2} \cdot \frac{1}{3}\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \left({t}^{2} \cdot \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({t}^{2}\right), \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(t \cdot t\right), \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \mathsf{*.f64}\left(2, \left({t}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \mathsf{*.f64}\left(2, \left(t \cdot t\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      18. *-lowering-*.f64 58.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

   9. Simplified 58.5%

      \[\leadsto \frac{\frac{2}{t}}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot 0.3333333333333333\right) + 2 \cdot \left(t \cdot t\right)\right)}}{\ell \cdot \ell}} \]
   10. Step-by-step derivation

       1. times-frac N/A

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

       2. associate-/r* N/A

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

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{t}}{\frac{k \cdot k}{\ell}}\right), \color{blue}{\left(\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + 2 \cdot \left(t \cdot t\right)}{\ell}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{t}\right), \left(\frac{k \cdot k}{\ell}\right)\right), \left(\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + 2 \cdot \left(t \cdot t\right)}}{\ell}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{k \cdot k}{\ell}\right)\right), \left(\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right)} + 2 \cdot \left(t \cdot t\right)}{\ell}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(k \cdot k\right), \ell\right)\right), \left(\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + \color{blue}{2 \cdot \left(t \cdot t\right)}}{\ell}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right)\right), \left(\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + \color{blue}{2} \cdot \left(t \cdot t\right)}{\ell}\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right)\right), \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + 2 \cdot \left(t \cdot t\right)\right), \color{blue}{\ell}\right)\right) \]

   11. Applied egg-rr 67.4%

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

   if 9.9999999999999993e-78 < t < 6.40000000000000011e152

   1. Initial program 72.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 72.3%

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

   5. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 73.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   7. Simplified 73.4%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right), \color{blue}{\left(\frac{\ell}{k}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\color{blue}{\ell}}{k}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      8. /-lowering-/.f64 76.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{k}\right)\right) \]

   9. Applied egg-rr 76.7%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot t\right) \cdot \left(t \cdot t\right)}\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{k \cdot t}}{t \cdot t}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, k\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot t}\right), \left(t \cdot t\right)\right), \mathsf{/.f64}\left(\color{blue}{\ell}, k\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right), \left(t \cdot t\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right), \left(t \cdot t\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, k\right)\right), \left(t \cdot t\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       7. *-lowering-*.f64 84.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, k\right)\right), \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

   11. Applied egg-rr 84.3%

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

   if 6.40000000000000011e152 < t

   1. Initial program 59.5%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 51.4%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{t}\right), \color{blue}{\left(\left(t \cdot t\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   6. Applied egg-rr 51.5%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{t \cdot \left(t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right)}{\color{blue}{\ell} \cdot \ell}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{t}{\ell} \cdot \color{blue}{\frac{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \color{blue}{\left(\frac{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}\right)}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\color{blue}{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}}{\ell}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right), \color{blue}{\ell}\right)\right)\right) \]

   8. Applied egg-rr 74.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right) \cdot t\right), \ell\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot t\right), \ell\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot t\right)\right), \ell\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \sin k\right), \left(\tan k \cdot t\right)\right), \ell\right)\right)\right) \]

   10. Applied egg-rr 87.5%

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

   11. Taylor expanded in k around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{k}, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(k, t\right), k\right), t\right)\right)\right), \mathsf{*.f64}\left(\mathsf{tan.f64}\left(k\right), t\right)\right), \ell\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 83.1%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-77}:\\ \;\;\;\;\frac{\frac{\frac{2}{t}}{\frac{k \cdot k}{\ell}}}{\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot 0.3333333333333333\right) + 2 \cdot \left(t \cdot t\right)}{\ell}}\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k}}{t \cdot t} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t}}{\frac{t}{\ell} \cdot \frac{\left(k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right) \cdot \left(t \cdot \tan k\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.9% accurate, 4.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{1}{t\_m \cdot t\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m}}{\frac{k \cdot k}{\ell}}}{\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t\_m \cdot t\_m\right) \cdot 0.3333333333333333\right) + 2 \cdot \left(t\_m \cdot t\_m\right)}{\ell}}\\ \mathbf{elif}\;t\_m \leq 1.7 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot k}}{t\_m \cdot t\_m} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t\_m}}{\frac{t\_m}{\ell} \cdot \frac{k \cdot \left(k \cdot \left(t\_m \cdot 2 + k \cdot \left(k \cdot \left(t\_m \cdot \left(0.3333333333333333 + t\_2\right) + \left(k \cdot k\right) \cdot \left(t\_m \cdot \left(\left(0.17222222222222222 + t\_2 \cdot -0.16666666666666666\right) + \frac{0.3333333333333333}{t\_m \cdot t\_m}\right) + \left(\left(0.0630952380952381 + t\_2 \cdot -0.05555555555555555\right) + \left(\left(\frac{0.13333333333333333}{t\_m \cdot t\_m} + 0.005555555555555556\right) + t\_2 \cdot 0.008333333333333333\right)\right) \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)\right)\right)\right)\right)}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ 1.0 (* t_m t_m))))
   (*
    t_s
    (if (<= t_m 7e-74)
      (/
       (/ (/ 2.0 t_m) (/ (* k k) l))
       (/
        (+
         (* (* k k) (+ 1.0 (* (* t_m t_m) 0.3333333333333333)))
         (* 2.0 (* t_m t_m)))
        l))
      (if (<= t_m 1.7e+152)
        (* (/ (/ l (* t_m k)) (* t_m t_m)) (/ l k))
        (/
         (/ 2.0 t_m)
         (*
          (/ t_m l)
          (/
           (*
            k
            (*
             k
             (+
              (* t_m 2.0)
              (*
               k
               (*
                k
                (+
                 (* t_m (+ 0.3333333333333333 t_2))
                 (*
                  (* k k)
                  (+
                   (*
                    t_m
                    (+
                     (+ 0.17222222222222222 (* t_2 -0.16666666666666666))
                     (/ 0.3333333333333333 (* t_m t_m))))
                   (*
                    (+
                     (+ 0.0630952380952381 (* t_2 -0.05555555555555555))
                     (+
                      (+
                       (/ 0.13333333333333333 (* t_m t_m))
                       0.005555555555555556)
                      (* t_2 0.008333333333333333)))
                    (* t_m (* k 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 t_2 = 1.0 / (t_m * t_m);
	double tmp;
	if (t_m <= 7e-74) {
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l);
	} else if (t_m <= 1.7e+152) {
		tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / k);
	} else {
		tmp = (2.0 / t_m) / ((t_m / l) * ((k * (k * ((t_m * 2.0) + (k * (k * ((t_m * (0.3333333333333333 + t_2)) + ((k * k) * ((t_m * ((0.17222222222222222 + (t_2 * -0.16666666666666666)) + (0.3333333333333333 / (t_m * t_m)))) + (((0.0630952380952381 + (t_2 * -0.05555555555555555)) + (((0.13333333333333333 / (t_m * t_m)) + 0.005555555555555556) + (t_2 * 0.008333333333333333))) * (t_m * (k * k))))))))))) / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 1.0d0 / (t_m * t_m)
    if (t_m <= 7d-74) then
        tmp = ((2.0d0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0d0 + ((t_m * t_m) * 0.3333333333333333d0))) + (2.0d0 * (t_m * t_m))) / l)
    else if (t_m <= 1.7d+152) then
        tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / k)
    else
        tmp = (2.0d0 / t_m) / ((t_m / l) * ((k * (k * ((t_m * 2.0d0) + (k * (k * ((t_m * (0.3333333333333333d0 + t_2)) + ((k * k) * ((t_m * ((0.17222222222222222d0 + (t_2 * (-0.16666666666666666d0))) + (0.3333333333333333d0 / (t_m * t_m)))) + (((0.0630952380952381d0 + (t_2 * (-0.05555555555555555d0))) + (((0.13333333333333333d0 / (t_m * t_m)) + 0.005555555555555556d0) + (t_2 * 0.008333333333333333d0))) * (t_m * (k * k))))))))))) / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = 1.0 / (t_m * t_m);
	double tmp;
	if (t_m <= 7e-74) {
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l);
	} else if (t_m <= 1.7e+152) {
		tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / k);
	} else {
		tmp = (2.0 / t_m) / ((t_m / l) * ((k * (k * ((t_m * 2.0) + (k * (k * ((t_m * (0.3333333333333333 + t_2)) + ((k * k) * ((t_m * ((0.17222222222222222 + (t_2 * -0.16666666666666666)) + (0.3333333333333333 / (t_m * t_m)))) + (((0.0630952380952381 + (t_2 * -0.05555555555555555)) + (((0.13333333333333333 / (t_m * t_m)) + 0.005555555555555556) + (t_2 * 0.008333333333333333))) * (t_m * (k * k))))))))))) / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = 1.0 / (t_m * t_m)
	tmp = 0
	if t_m <= 7e-74:
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l)
	elif t_m <= 1.7e+152:
		tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / k)
	else:
		tmp = (2.0 / t_m) / ((t_m / l) * ((k * (k * ((t_m * 2.0) + (k * (k * ((t_m * (0.3333333333333333 + t_2)) + ((k * k) * ((t_m * ((0.17222222222222222 + (t_2 * -0.16666666666666666)) + (0.3333333333333333 / (t_m * t_m)))) + (((0.0630952380952381 + (t_2 * -0.05555555555555555)) + (((0.13333333333333333 / (t_m * t_m)) + 0.005555555555555556) + (t_2 * 0.008333333333333333))) * (t_m * (k * 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)
	t_2 = Float64(1.0 / Float64(t_m * t_m))
	tmp = 0.0
	if (t_m <= 7e-74)
		tmp = Float64(Float64(Float64(2.0 / t_m) / Float64(Float64(k * k) / l)) / Float64(Float64(Float64(Float64(k * k) * Float64(1.0 + Float64(Float64(t_m * t_m) * 0.3333333333333333))) + Float64(2.0 * Float64(t_m * t_m))) / l));
	elseif (t_m <= 1.7e+152)
		tmp = Float64(Float64(Float64(l / Float64(t_m * k)) / Float64(t_m * t_m)) * Float64(l / k));
	else
		tmp = Float64(Float64(2.0 / t_m) / Float64(Float64(t_m / l) * Float64(Float64(k * Float64(k * Float64(Float64(t_m * 2.0) + Float64(k * Float64(k * Float64(Float64(t_m * Float64(0.3333333333333333 + t_2)) + Float64(Float64(k * k) * Float64(Float64(t_m * Float64(Float64(0.17222222222222222 + Float64(t_2 * -0.16666666666666666)) + Float64(0.3333333333333333 / Float64(t_m * t_m)))) + Float64(Float64(Float64(0.0630952380952381 + Float64(t_2 * -0.05555555555555555)) + Float64(Float64(Float64(0.13333333333333333 / Float64(t_m * t_m)) + 0.005555555555555556) + Float64(t_2 * 0.008333333333333333))) * Float64(t_m * Float64(k * k))))))))))) / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = 1.0 / (t_m * t_m);
	tmp = 0.0;
	if (t_m <= 7e-74)
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l);
	elseif (t_m <= 1.7e+152)
		tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / k);
	else
		tmp = (2.0 / t_m) / ((t_m / l) * ((k * (k * ((t_m * 2.0) + (k * (k * ((t_m * (0.3333333333333333 + t_2)) + ((k * k) * ((t_m * ((0.17222222222222222 + (t_2 * -0.16666666666666666)) + (0.3333333333333333 / (t_m * t_m)))) + (((0.0630952380952381 + (t_2 * -0.05555555555555555)) + (((0.13333333333333333 / (t_m * t_m)) + 0.005555555555555556) + (t_2 * 0.008333333333333333))) * (t_m * (k * k))))))))))) / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(1.0 / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7e-74], N[(N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(k * k), $MachinePrecision] * N[(1.0 + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.7e+152], N[(N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(k * N[(k * N[(N[(t$95$m * 2.0), $MachinePrecision] + N[(k * N[(k * N[(N[(t$95$m * N[(0.3333333333333333 + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * N[(N[(0.17222222222222222 + N[(t$95$2 * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.0630952380952381 + N[(t$95$2 * -0.05555555555555555), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.13333333333333333 / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 0.005555555555555556), $MachinePrecision] + N[(t$95$2 * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 7.00000000000000029e-74

   1. Initial program 52.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 47.5%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{t}\right), \color{blue}{\left(\left(t \cdot t\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   6. Applied egg-rr 51.4%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\color{blue}{\left({k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\ell}, \ell\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right) + 2 \cdot {t}^{2}\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left({k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({t}^{2} \cdot \left(\frac{1}{{t}^{2}} + \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({t}^{2} \cdot \frac{1}{{t}^{2}} + {t}^{2} \cdot \frac{1}{3}\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      11. rgt-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(1 + {t}^{2} \cdot \frac{1}{3}\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \left({t}^{2} \cdot \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({t}^{2}\right), \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(t \cdot t\right), \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \mathsf{*.f64}\left(2, \left({t}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \mathsf{*.f64}\left(2, \left(t \cdot t\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      18. *-lowering-*.f64 58.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

   9. Simplified 58.5%

      \[\leadsto \frac{\frac{2}{t}}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot 0.3333333333333333\right) + 2 \cdot \left(t \cdot t\right)\right)}}{\ell \cdot \ell}} \]
   10. Step-by-step derivation

       1. times-frac N/A

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

       2. associate-/r* N/A

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

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{t}}{\frac{k \cdot k}{\ell}}\right), \color{blue}{\left(\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + 2 \cdot \left(t \cdot t\right)}{\ell}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{t}\right), \left(\frac{k \cdot k}{\ell}\right)\right), \left(\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + 2 \cdot \left(t \cdot t\right)}}{\ell}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{k \cdot k}{\ell}\right)\right), \left(\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right)} + 2 \cdot \left(t \cdot t\right)}{\ell}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(k \cdot k\right), \ell\right)\right), \left(\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + \color{blue}{2 \cdot \left(t \cdot t\right)}}{\ell}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right)\right), \left(\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + \color{blue}{2} \cdot \left(t \cdot t\right)}{\ell}\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right)\right), \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + 2 \cdot \left(t \cdot t\right)\right), \color{blue}{\ell}\right)\right) \]

   11. Applied egg-rr 67.4%

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

   if 7.00000000000000029e-74 < t < 1.7000000000000001e152

   1. Initial program 72.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 72.3%

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

   5. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 73.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   7. Simplified 73.4%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right), \color{blue}{\left(\frac{\ell}{k}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\color{blue}{\ell}}{k}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      8. /-lowering-/.f64 76.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{k}\right)\right) \]

   9. Applied egg-rr 76.7%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot t\right) \cdot \left(t \cdot t\right)}\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{k \cdot t}}{t \cdot t}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, k\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot t}\right), \left(t \cdot t\right)\right), \mathsf{/.f64}\left(\color{blue}{\ell}, k\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right), \left(t \cdot t\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right), \left(t \cdot t\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, k\right)\right), \left(t \cdot t\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       7. *-lowering-*.f64 84.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, k\right)\right), \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

   11. Applied egg-rr 84.3%

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

   if 1.7000000000000001e152 < t

   1. Initial program 59.5%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 51.4%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{t}\right), \color{blue}{\left(\left(t \cdot t\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   6. Applied egg-rr 51.5%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{t \cdot \left(t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right)}{\color{blue}{\ell} \cdot \ell}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{t}{\ell} \cdot \color{blue}{\frac{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \color{blue}{\left(\frac{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}\right)}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\color{blue}{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}}{\ell}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right), \color{blue}{\ell}\right)\right)\right) \]

   8. Applied egg-rr 74.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right) \cdot t\right), \ell\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot t\right), \ell\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot t\right)\right), \ell\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(2 + \frac{k}{\frac{t \cdot t}{k}}\right) \cdot \sin k\right), \left(\tan k \cdot t\right)\right), \ell\right)\right)\right) \]

   10. Applied egg-rr 87.5%

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

   11. Taylor expanded in k around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\color{blue}{\left({k}^{2} \cdot \left(2 \cdot t + {k}^{2} \cdot \left(t \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right) + {k}^{2} \cdot \left(t \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right) + {k}^{2} \cdot \left(t \cdot \left(\frac{271}{2520} + \left(\frac{-1}{6} \cdot \left(\frac{4}{15} + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right) + \left(\frac{1}{120} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{2}{15} \cdot \frac{1}{{t}^{2}}\right)\right)\right)\right)\right)\right)\right)\right)}, \ell\right)\right)\right) \]

   13. Simplified 83.0%

       \[\leadsto \frac{\frac{2}{t}}{\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot \left(k \cdot \left(2 \cdot t + k \cdot \left(k \cdot \left(t \cdot \left(0.3333333333333333 + \frac{1}{t \cdot t}\right) + \left(k \cdot k\right) \cdot \left(t \cdot \left(\left(0.17222222222222222 + -0.16666666666666666 \cdot \frac{1}{t \cdot t}\right) + \frac{0.3333333333333333}{t \cdot t}\right) + \left(\left(0.0630952380952381 + -0.05555555555555555 \cdot \frac{1}{t \cdot t}\right) + \left(\left(\frac{0.13333333333333333}{t \cdot t} + 0.005555555555555556\right) + 0.008333333333333333 \cdot \frac{1}{t \cdot t}\right)\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right)\right)\right)\right)}}{\ell}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{\frac{2}{t}}{\frac{k \cdot k}{\ell}}}{\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot 0.3333333333333333\right) + 2 \cdot \left(t \cdot t\right)}{\ell}}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k}}{t \cdot t} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t}}{\frac{t}{\ell} \cdot \frac{k \cdot \left(k \cdot \left(t \cdot 2 + k \cdot \left(k \cdot \left(t \cdot \left(0.3333333333333333 + \frac{1}{t \cdot t}\right) + \left(k \cdot k\right) \cdot \left(t \cdot \left(\left(0.17222222222222222 + \frac{1}{t \cdot t} \cdot -0.16666666666666666\right) + \frac{0.3333333333333333}{t \cdot t}\right) + \left(\left(0.0630952380952381 + \frac{1}{t \cdot t} \cdot -0.05555555555555555\right) + \left(\left(\frac{0.13333333333333333}{t \cdot t} + 0.005555555555555556\right) + \frac{1}{t \cdot t} \cdot 0.008333333333333333\right)\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right)\right)\right)\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.0% accurate, 12.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m}}{\frac{k \cdot k}{\ell}}}{\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t\_m \cdot t\_m\right) \cdot 0.3333333333333333\right) + 2 \cdot \left(t\_m \cdot t\_m\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot k}}{t\_m \cdot t\_m} \cdot \frac{\ell}{k}\\ \end{array} \end{array} \]

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 4.8e-78)
    (/
     (/ (/ 2.0 t_m) (/ (* k k) l))
     (/
      (+
       (* (* k k) (+ 1.0 (* (* t_m t_m) 0.3333333333333333)))
       (* 2.0 (* t_m t_m)))
      l))
    (* (/ (/ l (* t_m k)) (* t_m t_m)) (/ l 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 <= 4.8e-78) {
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l);
	} else {
		tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / 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 <= 4.8d-78) then
        tmp = ((2.0d0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0d0 + ((t_m * t_m) * 0.3333333333333333d0))) + (2.0d0 * (t_m * t_m))) / l)
    else
        tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / 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 <= 4.8e-78) {
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l);
	} else {
		tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / 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 <= 4.8e-78:
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l)
	else:
		tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / 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 <= 4.8e-78)
		tmp = Float64(Float64(Float64(2.0 / t_m) / Float64(Float64(k * k) / l)) / Float64(Float64(Float64(Float64(k * k) * Float64(1.0 + Float64(Float64(t_m * t_m) * 0.3333333333333333))) + Float64(2.0 * Float64(t_m * t_m))) / l));
	else
		tmp = Float64(Float64(Float64(l / Float64(t_m * k)) / Float64(t_m * t_m)) * Float64(l / 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 <= 4.8e-78)
		tmp = ((2.0 / t_m) / ((k * k) / l)) / ((((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m))) / l);
	else
		tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / 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, 4.8e-78], N[(N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(k * k), $MachinePrecision] * N[(1.0 + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.79999999999999999e-78

   1. Initial program 52.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 47.5%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{t}\right), \color{blue}{\left(\left(t \cdot t\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   6. Applied egg-rr 51.4%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\color{blue}{\left({k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\ell}, \ell\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right) + 2 \cdot {t}^{2}\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left({k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({t}^{2} \cdot \left(\frac{1}{{t}^{2}} + \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({t}^{2} \cdot \frac{1}{{t}^{2}} + {t}^{2} \cdot \frac{1}{3}\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      11. rgt-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(1 + {t}^{2} \cdot \frac{1}{3}\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \left({t}^{2} \cdot \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({t}^{2}\right), \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(t \cdot t\right), \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \mathsf{*.f64}\left(2, \left({t}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \mathsf{*.f64}\left(2, \left(t \cdot t\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      18. *-lowering-*.f64 58.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

   9. Simplified 58.5%

      \[\leadsto \frac{\frac{2}{t}}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot 0.3333333333333333\right) + 2 \cdot \left(t \cdot t\right)\right)}}{\ell \cdot \ell}} \]
   10. Step-by-step derivation

       1. times-frac N/A

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

       2. associate-/r* N/A

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

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{t}}{\frac{k \cdot k}{\ell}}\right), \color{blue}{\left(\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + 2 \cdot \left(t \cdot t\right)}{\ell}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{t}\right), \left(\frac{k \cdot k}{\ell}\right)\right), \left(\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + 2 \cdot \left(t \cdot t\right)}}{\ell}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{k \cdot k}{\ell}\right)\right), \left(\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right)} + 2 \cdot \left(t \cdot t\right)}{\ell}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(k \cdot k\right), \ell\right)\right), \left(\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + \color{blue}{2 \cdot \left(t \cdot t\right)}}{\ell}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right)\right), \left(\frac{\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + \color{blue}{2} \cdot \left(t \cdot t\right)}{\ell}\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \ell\right)\right), \mathsf{/.f64}\left(\left(\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + 2 \cdot \left(t \cdot t\right)\right), \color{blue}{\ell}\right)\right) \]

   11. Applied egg-rr 67.4%

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

   if 4.79999999999999999e-78 < t

   1. Initial program 66.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 61.9%

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

   5. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 66.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   7. Simplified 66.5%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right), \color{blue}{\left(\frac{\ell}{k}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\color{blue}{\ell}}{k}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      8. /-lowering-/.f64 70.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{k}\right)\right) \]

   9. Applied egg-rr 70.9%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot t\right) \cdot \left(t \cdot t\right)}\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{k \cdot t}}{t \cdot t}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, k\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot t}\right), \left(t \cdot t\right)\right), \mathsf{/.f64}\left(\color{blue}{\ell}, k\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right), \left(t \cdot t\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right), \left(t \cdot t\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, k\right)\right), \left(t \cdot t\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       7. *-lowering-*.f64 74.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, k\right)\right), \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

   11. Applied egg-rr 74.7%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{t \cdot k}}{t \cdot t}} \cdot \frac{\ell}{k} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 68.9% accurate, 12.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot k}}{t\_m \cdot t\_m} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{2}{t\_m}}{\frac{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \left(1 + \left(t\_m \cdot t\_m\right) \cdot 0.3333333333333333\right) + 2 \cdot \left(t\_m \cdot t\_m\right)\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.2e-132)
    (* (/ (/ l (* t_m k)) (* t_m t_m)) (/ l k))
    (*
     l
     (/
      (/ 2.0 t_m)
      (/
       (*
        (* k k)
        (+
         (* (* k k) (+ 1.0 (* (* t_m t_m) 0.3333333333333333)))
         (* 2.0 (* t_m t_m))))
       l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.2e-132) {
		tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / k);
	} else {
		tmp = l * ((2.0 / t_m) / (((k * k) * (((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m)))) / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.2d-132) then
        tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / k)
    else
        tmp = l * ((2.0d0 / t_m) / (((k * k) * (((k * k) * (1.0d0 + ((t_m * t_m) * 0.3333333333333333d0))) + (2.0d0 * (t_m * t_m)))) / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.2e-132) {
		tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / k);
	} else {
		tmp = l * ((2.0 / t_m) / (((k * k) * (((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m)))) / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.2e-132:
		tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / k)
	else:
		tmp = l * ((2.0 / t_m) / (((k * k) * (((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m)))) / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.2e-132)
		tmp = Float64(Float64(Float64(l / Float64(t_m * k)) / Float64(t_m * t_m)) * Float64(l / k));
	else
		tmp = Float64(l * Float64(Float64(2.0 / t_m) / Float64(Float64(Float64(k * k) * Float64(Float64(Float64(k * k) * Float64(1.0 + Float64(Float64(t_m * t_m) * 0.3333333333333333))) + Float64(2.0 * Float64(t_m * t_m)))) / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.2e-132)
		tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / k);
	else
		tmp = l * ((2.0 / t_m) / (((k * k) * (((k * k) * (1.0 + ((t_m * t_m) * 0.3333333333333333))) + (2.0 * (t_m * t_m)))) / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.2e-132], N[(N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] * N[(1.0 + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.20000000000000008e-132

   1. Initial program 57.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 50.7%

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

   5. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 57.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   7. Simplified 57.9%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right), \color{blue}{\left(\frac{\ell}{k}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\color{blue}{\ell}}{k}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      8. /-lowering-/.f64 68.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{k}\right)\right) \]

   9. Applied egg-rr 68.1%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot t\right) \cdot \left(t \cdot t\right)}\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{k \cdot t}}{t \cdot t}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, k\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot t}\right), \left(t \cdot t\right)\right), \mathsf{/.f64}\left(\color{blue}{\ell}, k\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right), \left(t \cdot t\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right), \left(t \cdot t\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, k\right)\right), \left(t \cdot t\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       7. *-lowering-*.f64 72.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, k\right)\right), \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

   11. Applied egg-rr 72.6%

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

   if 1.20000000000000008e-132 < k

   1. Initial program 54.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 54.1%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{t}\right), \color{blue}{\left(\left(t \cdot t\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   6. Applied egg-rr 60.0%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\color{blue}{\left({k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right)}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\ell}, \ell\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right) + 2 \cdot {t}^{2}\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left({k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({t}^{2} \cdot \left(\frac{1}{{t}^{2}} + \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({t}^{2} \cdot \frac{1}{{t}^{2}} + {t}^{2} \cdot \frac{1}{3}\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      11. rgt-mult-inverse N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(1 + {t}^{2} \cdot \frac{1}{3}\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \left({t}^{2} \cdot \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({t}^{2}\right), \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(t \cdot t\right), \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \left(2 \cdot {t}^{2}\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \mathsf{*.f64}\left(2, \left({t}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \mathsf{*.f64}\left(2, \left(t \cdot t\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

      18. *-lowering-*.f64 66.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right) \]

   9. Simplified 66.7%

      \[\leadsto \frac{\frac{2}{t}}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot 0.3333333333333333\right) + 2 \cdot \left(t \cdot t\right)\right)}}{\ell \cdot \ell}} \]
   10. Step-by-step derivation

       1. associate-/r* N/A

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

       2. associate-/r/ N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{t}}{\frac{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot \frac{1}{3}\right) + 2 \cdot \left(t \cdot t\right)\right)}{\ell}}\right), \color{blue}{\ell}\right) \]

   11. Applied egg-rr 72.6%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k}}{t \cdot t} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{2}{t}}{\frac{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \left(1 + \left(t \cdot t\right) \cdot 0.3333333333333333\right) + 2 \cdot \left(t \cdot t\right)\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.7% accurate, 19.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot k}}{t\_m \cdot t\_m} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t\_m}}{\frac{t\_m}{\ell} \cdot \frac{2 \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 5e-133)
    (* (/ (/ l (* t_m k)) (* t_m t_m)) (/ l k))
    (/ (/ 2.0 t_m) (* (/ t_m l) (/ (* 2.0 (* t_m (* k 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 (k <= 5e-133) {
		tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / k);
	} else {
		tmp = (2.0 / t_m) / ((t_m / l) * ((2.0 * (t_m * (k * k))) / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5d-133) then
        tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / k)
    else
        tmp = (2.0d0 / t_m) / ((t_m / l) * ((2.0d0 * (t_m * (k * k))) / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5e-133) {
		tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / k);
	} else {
		tmp = (2.0 / t_m) / ((t_m / l) * ((2.0 * (t_m * (k * k))) / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 5e-133:
		tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / k)
	else:
		tmp = (2.0 / t_m) / ((t_m / l) * ((2.0 * (t_m * (k * 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 (k <= 5e-133)
		tmp = Float64(Float64(Float64(l / Float64(t_m * k)) / Float64(t_m * t_m)) * Float64(l / k));
	else
		tmp = Float64(Float64(2.0 / t_m) / Float64(Float64(t_m / l) * Float64(Float64(2.0 * Float64(t_m * Float64(k * k))) / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 5e-133)
		tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / k);
	else
		tmp = (2.0 / t_m) / ((t_m / l) * ((2.0 * (t_m * (k * k))) / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 5e-133], N[(N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(2.0 * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.9999999999999999e-133

   1. Initial program 57.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 50.7%

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

   5. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 57.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   7. Simplified 57.9%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right), \color{blue}{\left(\frac{\ell}{k}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\color{blue}{\ell}}{k}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      8. /-lowering-/.f64 68.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{k}\right)\right) \]

   9. Applied egg-rr 68.1%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot t\right) \cdot \left(t \cdot t\right)}\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{k \cdot t}}{t \cdot t}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, k\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot t}\right), \left(t \cdot t\right)\right), \mathsf{/.f64}\left(\color{blue}{\ell}, k\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right), \left(t \cdot t\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right), \left(t \cdot t\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, k\right)\right), \left(t \cdot t\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       7. *-lowering-*.f64 72.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, k\right)\right), \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

   11. Applied egg-rr 72.6%

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

   if 4.9999999999999999e-133 < k

   1. Initial program 54.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 54.1%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{t}\right), \color{blue}{\left(\left(t \cdot t\right) \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)}{\ell \cdot \ell}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   6. Applied egg-rr 60.0%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{t \cdot \left(t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right)}{\color{blue}{\ell} \cdot \ell}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \left(\frac{t}{\ell} \cdot \color{blue}{\frac{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \color{blue}{\left(\frac{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}\right)}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\color{blue}{t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}}{\ell}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(t \cdot \left(\left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right), \color{blue}{\ell}\right)\right)\right) \]

   8. Applied egg-rr 72.2%

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

   9. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \left({k}^{2} \cdot t\right)\right)}, \ell\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({k}^{2} \cdot t\right)\right), \ell\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2}\right), t\right)\right), \ell\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right), \ell\right)\right)\right) \]

       4. *-lowering-*.f64 72.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \ell\right)\right)\right) \]

   11. Simplified 72.2%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k}}{t \cdot t} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t}}{\frac{t}{\ell} \cdot \frac{2 \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.8% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot k}}{t\_m \cdot t\_m} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

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 2e-133)
    (* (/ (/ l (* t_m k)) (* t_m t_m)) (/ l 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 <= 2e-133) {
		tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / 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 <= 2d-133) then
        tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / 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 <= 2e-133) {
		tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / 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 <= 2e-133:
		tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / 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 <= 2e-133)
		tmp = Float64(Float64(Float64(l / Float64(t_m * k)) / Float64(t_m * t_m)) * Float64(l / 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 <= 2e-133)
		tmp = ((l / (t_m * k)) / (t_m * t_m)) * (l / 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, 2e-133], N[(N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k), $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 < 2.0000000000000001e-133

   1. Initial program 57.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 50.7%

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

   5. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 57.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   7. Simplified 57.9%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right), \color{blue}{\left(\frac{\ell}{k}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\color{blue}{\ell}}{k}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      8. /-lowering-/.f64 68.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{k}\right)\right) \]

   9. Applied egg-rr 68.1%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\left(k \cdot t\right) \cdot \left(t \cdot t\right)}\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{k \cdot t}}{t \cdot t}\right), \mathsf{/.f64}\left(\color{blue}{\ell}, k\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot t}\right), \left(t \cdot t\right)\right), \mathsf{/.f64}\left(\color{blue}{\ell}, k\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot t\right)\right), \left(t \cdot t\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right), \left(t \cdot t\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, k\right)\right), \left(t \cdot t\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

       7. *-lowering-*.f64 72.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, k\right)\right), \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\ell, k\right)\right) \]

   11. Applied egg-rr 72.6%

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

   if 2.0000000000000001e-133 < k

   1. Initial program 54.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 54.1%

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

   5. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 57.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   7. Simplified 57.6%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right), \color{blue}{\left(\frac{\ell}{k}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\color{blue}{\ell}}{k}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      8. /-lowering-/.f64 58.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{k}\right)\right) \]

   9. Applied egg-rr 58.5%

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

       1. frac-times N/A

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

       2. *-commutative N/A

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

       3. associate-*r* 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. associate-/l* N/A

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

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot \color{blue}{k}\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right) \cdot k\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right)\right) \]

       13. *-lowering-*.f64 59.4%

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 59.4%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot t\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right), \color{blue}{t}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right), t\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right), t\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right), t\right)\right)\right) \]

       9. *-lowering-*.f64 64.9%

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right), t\right)\right)\right) \]

   13. Applied egg-rr 64.9%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k}}{t \cdot t} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.8% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-89}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{k \cdot \left(t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

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 1e-89)
    (* l (/ l (* t_m (* t_m (* t_m (* k k))))))
    (* l (/ l (* k (* t_m (* 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 <= 1e-89) {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	} else {
		tmp = l * (l / (k * (t_m * (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 <= 1d-89) then
        tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))))
    else
        tmp = l * (l / (k * (t_m * (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 <= 1e-89) {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	} else {
		tmp = l * (l / (k * (t_m * (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 <= 1e-89:
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))))
	else:
		tmp = l * (l / (k * (t_m * (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 <= 1e-89)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k))))));
	else
		tmp = Float64(l * Float64(l / Float64(k * Float64(t_m * 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 <= 1e-89)
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	else
		tmp = l * (l / (k * (t_m * (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, 1e-89], 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], N[(l * N[(l / N[(k * N[(t$95$m * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.00000000000000004e-89

   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. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 47.7%

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

   5. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 54.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   7. Simplified 54.3%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right), \color{blue}{\left(\frac{\ell}{k}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\color{blue}{\ell}}{k}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      8. /-lowering-/.f64 62.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{k}\right)\right) \]

   9. Applied egg-rr 62.4%

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

       1. frac-times N/A

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

       2. *-commutative N/A

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

       3. associate-*r* 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. associate-/l* N/A

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

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot \color{blue}{k}\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right) \cdot k\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right)\right) \]

       13. *-lowering-*.f64 61.0%

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 61.0%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot t\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right), \color{blue}{t}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right), t\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right), t\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right), t\right)\right)\right) \]

       9. *-lowering-*.f64 62.3%

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right), t\right)\right)\right) \]

   13. Applied egg-rr 62.3%

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

   if 1.00000000000000004e-89 < t

   1. Initial program 65.3%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

   3. Simplified 61.2%

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

   5. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 65.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   7. Simplified 65.7%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right), \color{blue}{\left(\frac{\ell}{k}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\color{blue}{\ell}}{k}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      8. /-lowering-/.f64 70.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{k}\right)\right) \]

   9. Applied egg-rr 70.0%

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

       1. frac-times N/A

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

       2. *-commutative N/A

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

       3. associate-*r* 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. associate-/l* N/A

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

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot \color{blue}{k}\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right) \cdot k\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right)\right) \]

       13. *-lowering-*.f64 70.0%

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 70.0%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(\left(k \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(\left(\left(k \cdot t\right) \cdot t\right) \cdot \color{blue}{t}\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\left(\left(k \cdot t\right) \cdot t\right), \color{blue}{t}\right)\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\left(\left(t \cdot k\right) \cdot t\right), t\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(t \cdot k\right), t\right), t\right)\right)\right)\right) \]

       6. *-lowering-*.f64 74.8%

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), t\right), t\right)\right)\right)\right) \]

   13. Applied egg-rr 74.8%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-89}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.1% accurate, 32.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right)}\right) \end{array} \]

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 (* k (* t_m (* 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 / (k * (t_m * (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 / (k * (t_m * (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 / (k * (t_m * (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 / (k * (t_m * (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(k * Float64(t_m * 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 / (k * (t_m * (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[(k * N[(t$95$m * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.3%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

   4. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

   7. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

   10. associate-*l/ N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

3. Simplified 51.9%

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

5. Taylor expanded in k around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

   8. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 57.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

7. Simplified 57.8%

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

   1. *-commutative N/A

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

   2. times-frac N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right), \color{blue}{\left(\frac{\ell}{k}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\color{blue}{\ell}}{k}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

   8. /-lowering-/.f64 64.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{k}\right)\right) \]

9. Applied egg-rr 64.8%

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

    1. frac-times N/A

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

    2. *-commutative N/A

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

    3. associate-*r* 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. associate-/l* N/A

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

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

    7. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot \color{blue}{k}\right)\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right) \cdot k\right)\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right)\right) \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right)\right) \]

    12. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right)\right) \]

    13. *-lowering-*.f64 63.8%

        \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right)\right) \]

11. Applied egg-rr 63.8%

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

    1. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(\left(k \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

    2. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(\left(\left(k \cdot t\right) \cdot t\right) \cdot \color{blue}{t}\right)\right)\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\left(\left(k \cdot t\right) \cdot t\right), \color{blue}{t}\right)\right)\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\left(\left(t \cdot k\right) \cdot t\right), t\right)\right)\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(t \cdot k\right), t\right), t\right)\right)\right)\right) \]

    6. *-lowering-*.f64 66.5%

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), t\right), t\right)\right)\right)\right) \]

13. Applied egg-rr 66.5%

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

14. Final simplification 66.5%

    \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)} \]
15. Add Preprocessing

Alternative 15: 60.1% accurate, 32.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right)\right)}\right) \end{array} \]

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 (* k (* k (* t_m (* 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 / (k * (k * (t_m * (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 / (k * (k * (t_m * (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 / (k * (k * (t_m * (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 / (k * (k * (t_m * (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(k * Float64(k * Float64(t_m * 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 / (k * (k * (t_m * (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[(k * N[(k * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.3%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

   4. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({t}^{3} \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

   7. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

   10. associate-*l/ N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(\frac{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{/.f64}\left(\left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \]

3. Simplified 51.9%

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

5. Taylor expanded in k around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

   8. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 57.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

7. Simplified 57.8%

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

   1. *-commutative N/A

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

   2. times-frac N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right), \color{blue}{\left(\frac{\ell}{k}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\color{blue}{\ell}}{k}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

   8. /-lowering-/.f64 64.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{k}\right)\right) \]

9. Applied egg-rr 64.8%

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

    1. frac-times N/A

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

    2. *-commutative N/A

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

    3. associate-*r* 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. associate-/l* N/A

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

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

    7. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot \color{blue}{k}\right)\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right) \cdot k\right)\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right)\right) \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\right)\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)\right)\right)\right) \]

    12. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right)\right) \]

    13. *-lowering-*.f64 63.8%

        \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right)\right) \]

11. Applied egg-rr 63.8%

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

Reproduce

herbie shell --seed 2024152 
(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))))