Toniolo and Linder, Equation (10+)

Percentage Accurate: 53.4% → 86.6%

Time: 19.1s

Alternatives: 15

Speedup: 23.4×

• 27.5% 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: 53.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{2 \cdot \ell}{t\_m \cdot \left(k \cdot k\right)} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{elif}\;t\_m \leq 1.1 \cdot 10^{+197}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{{t\_m}^{1.5}}{\ell} \cdot \frac{\sin k \cdot {t\_m}^{1.5}}{\ell}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\frac{\frac{\ell}{t\_m \cdot k}}{t\_m}}{k}\\ \end{array} \end{array} \]

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 2.2e-80)
    (* (/ (* 2.0 l) (* t_m (* k k))) (/ (/ l (sin k)) (tan k)))
    (if (<= t_m 1.1e+197)
      (/
       2.0
       (*
        (* (tan k) (* (/ (pow t_m 1.5) l) (/ (* (sin k) (pow t_m 1.5)) l)))
        (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))))
      (* (/ l t_m) (/ (/ (/ l (* t_m k)) t_m) k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.2e-80) {
		tmp = ((2.0 * l) / (t_m * (k * k))) * ((l / sin(k)) / tan(k));
	} else if (t_m <= 1.1e+197) {
		tmp = 2.0 / ((tan(k) * ((pow(t_m, 1.5) / l) * ((sin(k) * pow(t_m, 1.5)) / l))) * (1.0 + (1.0 + pow((k / t_m), 2.0))));
	} else {
		tmp = (l / t_m) * (((l / (t_m * k)) / 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 <= 2.2d-80) then
        tmp = ((2.0d0 * l) / (t_m * (k * k))) * ((l / sin(k)) / tan(k))
    else if (t_m <= 1.1d+197) then
        tmp = 2.0d0 / ((tan(k) * (((t_m ** 1.5d0) / l) * ((sin(k) * (t_m ** 1.5d0)) / l))) * (1.0d0 + (1.0d0 + ((k / t_m) ** 2.0d0))))
    else
        tmp = (l / t_m) * (((l / (t_m * k)) / 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 <= 2.2e-80) {
		tmp = ((2.0 * l) / (t_m * (k * k))) * ((l / Math.sin(k)) / Math.tan(k));
	} else if (t_m <= 1.1e+197) {
		tmp = 2.0 / ((Math.tan(k) * ((Math.pow(t_m, 1.5) / l) * ((Math.sin(k) * Math.pow(t_m, 1.5)) / l))) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0))));
	} else {
		tmp = (l / t_m) * (((l / (t_m * k)) / 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 <= 2.2e-80:
		tmp = ((2.0 * l) / (t_m * (k * k))) * ((l / math.sin(k)) / math.tan(k))
	elif t_m <= 1.1e+197:
		tmp = 2.0 / ((math.tan(k) * ((math.pow(t_m, 1.5) / l) * ((math.sin(k) * math.pow(t_m, 1.5)) / l))) * (1.0 + (1.0 + math.pow((k / t_m), 2.0))))
	else:
		tmp = (l / t_m) * (((l / (t_m * k)) / 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 <= 2.2e-80)
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(t_m * Float64(k * k))) * Float64(Float64(l / sin(k)) / tan(k)));
	elseif (t_m <= 1.1e+197)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64((t_m ^ 1.5) / l) * Float64(Float64(sin(k) * (t_m ^ 1.5)) / l))) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))));
	else
		tmp = Float64(Float64(l / t_m) * Float64(Float64(Float64(l / Float64(t_m * k)) / 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 <= 2.2e-80)
		tmp = ((2.0 * l) / (t_m * (k * k))) * ((l / sin(k)) / tan(k));
	elseif (t_m <= 1.1e+197)
		tmp = 2.0 / ((tan(k) * (((t_m ^ 1.5) / l) * ((sin(k) * (t_m ^ 1.5)) / l))) * (1.0 + (1.0 + ((k / t_m) ^ 2.0))));
	else
		tmp = (l / t_m) * (((l / (t_m * k)) / 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, 2.2e-80], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.1e+197], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.2000000000000001e-80

   1. Initial program 47.8%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. cube-unmult N/A

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

      4. associate-/r* N/A

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

      5. associate-/r* N/A

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

      6. associate-/r/ N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

   4. Applied egg-rr 42.4%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 70.7%

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

   7. Simplified 70.7%

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

   if 2.2000000000000001e-80 < t < 1.09999999999999995e197

   1. Initial program 77.9%

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

      1. cube-unmult N/A

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

      2. associate-*l/ N/A

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

      3. cube-unmult N/A

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

      4. sqr-pow N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

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

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

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

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

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

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

      10. metadata-eval N/A

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

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

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

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

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

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

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

      14. metadata-eval N/A

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

      15. sin-lowering-sin.f64 86.9%

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

   4. Applied egg-rr 86.9%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   if 1.09999999999999995e197 < t

   1. Initial program 45.9%

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

   3. Taylor expanded in k around 0

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

      1. /-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. *-commutative N/A

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

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

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 35.2%

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

   5. Simplified 35.2%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-lowering-*.f64 54.7%

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

   7. Applied egg-rr 54.7%

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

      1. associate-/r* N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 90.3%

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

   9. Applied egg-rr 90.3%

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

       1. associate-/l/ N/A

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

       2. associate-/r* N/A

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

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

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

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

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

       5. *-lowering-*.f64 90.5%

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

   11. Applied egg-rr 90.5%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+197}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \frac{\sin k \cdot {t}^{1.5}}{\ell}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{\frac{\ell}{t \cdot k}}{t}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{t\_m}^{1.5}}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.65 \cdot 10^{-78}:\\ \;\;\;\;\frac{2 \cdot \ell}{t\_m \cdot \left(k \cdot k\right)} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{elif}\;t\_m \leq 4.9 \cdot 10^{+173}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_2} \cdot \frac{\frac{1}{\sin k}}{t\_2}}{\tan k}}{2 + k \cdot \frac{k}{t\_m \cdot t\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\frac{\frac{\ell}{t\_m \cdot k}}{t\_m}}{k}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (pow t_m 1.5) l)))
   (*
    t_s
    (if (<= t_m 1.65e-78)
      (* (/ (* 2.0 l) (* t_m (* k k))) (/ (/ l (sin k)) (tan k)))
      (if (<= t_m 4.9e+173)
        (/
         (/ (* (/ 2.0 t_2) (/ (/ 1.0 (sin k)) t_2)) (tan k))
         (+ 2.0 (* k (/ k (* t_m t_m)))))
        (* (/ l t_m) (/ (/ (/ l (* t_m k)) t_m) k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(t_m, 1.5) / l;
	double tmp;
	if (t_m <= 1.65e-78) {
		tmp = ((2.0 * l) / (t_m * (k * k))) * ((l / sin(k)) / tan(k));
	} else if (t_m <= 4.9e+173) {
		tmp = (((2.0 / t_2) * ((1.0 / sin(k)) / t_2)) / tan(k)) / (2.0 + (k * (k / (t_m * t_m))));
	} else {
		tmp = (l / t_m) * (((l / (t_m * k)) / 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) :: t_2
    real(8) :: tmp
    t_2 = (t_m ** 1.5d0) / l
    if (t_m <= 1.65d-78) then
        tmp = ((2.0d0 * l) / (t_m * (k * k))) * ((l / sin(k)) / tan(k))
    else if (t_m <= 4.9d+173) then
        tmp = (((2.0d0 / t_2) * ((1.0d0 / sin(k)) / t_2)) / tan(k)) / (2.0d0 + (k * (k / (t_m * t_m))))
    else
        tmp = (l / t_m) * (((l / (t_m * k)) / 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 t_2 = Math.pow(t_m, 1.5) / l;
	double tmp;
	if (t_m <= 1.65e-78) {
		tmp = ((2.0 * l) / (t_m * (k * k))) * ((l / Math.sin(k)) / Math.tan(k));
	} else if (t_m <= 4.9e+173) {
		tmp = (((2.0 / t_2) * ((1.0 / Math.sin(k)) / t_2)) / Math.tan(k)) / (2.0 + (k * (k / (t_m * t_m))));
	} else {
		tmp = (l / t_m) * (((l / (t_m * k)) / 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):
	t_2 = math.pow(t_m, 1.5) / l
	tmp = 0
	if t_m <= 1.65e-78:
		tmp = ((2.0 * l) / (t_m * (k * k))) * ((l / math.sin(k)) / math.tan(k))
	elif t_m <= 4.9e+173:
		tmp = (((2.0 / t_2) * ((1.0 / math.sin(k)) / t_2)) / math.tan(k)) / (2.0 + (k * (k / (t_m * t_m))))
	else:
		tmp = (l / t_m) * (((l / (t_m * k)) / 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)
	t_2 = Float64((t_m ^ 1.5) / l)
	tmp = 0.0
	if (t_m <= 1.65e-78)
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(t_m * Float64(k * k))) * Float64(Float64(l / sin(k)) / tan(k)));
	elseif (t_m <= 4.9e+173)
		tmp = Float64(Float64(Float64(Float64(2.0 / t_2) * Float64(Float64(1.0 / sin(k)) / t_2)) / tan(k)) / Float64(2.0 + Float64(k * Float64(k / Float64(t_m * t_m)))));
	else
		tmp = Float64(Float64(l / t_m) * Float64(Float64(Float64(l / Float64(t_m * k)) / 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)
	t_2 = (t_m ^ 1.5) / l;
	tmp = 0.0;
	if (t_m <= 1.65e-78)
		tmp = ((2.0 * l) / (t_m * (k * k))) * ((l / sin(k)) / tan(k));
	elseif (t_m <= 4.9e+173)
		tmp = (((2.0 / t_2) * ((1.0 / sin(k)) / t_2)) / tan(k)) / (2.0 + (k * (k / (t_m * t_m))));
	else
		tmp = (l / t_m) * (((l / (t_m * k)) / 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_] := Block[{t$95$2 = N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.65e-78], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.9e+173], N[(N[(N[(N[(2.0 / t$95$2), $MachinePrecision] * N[(N[(1.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.64999999999999991e-78

   1. Initial program 47.8%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. cube-unmult N/A

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

      4. associate-/r* N/A

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

      5. associate-/r* N/A

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

      6. associate-/r/ N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

   4. Applied egg-rr 42.4%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 70.7%

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

   7. Simplified 70.7%

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

   if 1.64999999999999991e-78 < t < 4.9000000000000001e173

   1. Initial program 77.0%

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

      1. associate-/r* N/A

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

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

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

      3. associate-/r* N/A

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

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

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

      5. associate-/r* N/A

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

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

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

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

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

      8. associate-/r* N/A

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

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

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

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

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

      11. cube-mult N/A

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

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

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

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

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

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

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

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

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

   3. Simplified 77.2%

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

      1. div-inv N/A

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

      2. associate-*l/ N/A

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

      3. associate-/r* N/A

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

      4. cube-unmult N/A

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

      5. sqr-pow N/A

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

      6. times-frac N/A

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

      7. times-frac N/A

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

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

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

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

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

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

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

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

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

      12. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      17. pow-lowering-pow.f64 N/A

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

      18. metadata-eval 86.4%

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

   6. Applied egg-rr 86.4%

      \[\leadsto \frac{\frac{\color{blue}{\frac{2}{\frac{{t}^{1.5}}{\ell}} \cdot \frac{\frac{1}{\sin k}}{\frac{{t}^{1.5}}{\ell}}}}{\tan k}}{2 + k \cdot \frac{k}{t \cdot t}} \]

   if 4.9000000000000001e173 < t

   1. Initial program 50.8%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. *-commutative N/A

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

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

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 36.5%

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

   5. Simplified 36.5%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-lowering-*.f64 54.3%

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

   7. Applied egg-rr 54.3%

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

      1. associate-/r* N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 91.2%

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

   9. Applied egg-rr 91.2%

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

       1. associate-/l/ N/A

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

       2. associate-/r* N/A

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

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

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

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

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

       5. *-lowering-*.f64 91.4%

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

   11. Applied egg-rr 91.4%

       \[\leadsto \frac{\ell}{t} \cdot \frac{\color{blue}{\frac{\frac{\ell}{t \cdot k}}{t}}}{k} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 77.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 8.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot k}}{k \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{t\_m \cdot \left(k \cdot k\right)} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 8.2e-7)
    (/ (/ (/ l t_m) (* t_m k)) (* k (/ t_m l)))
    (* (/ (* 2.0 l) (* t_m (* k k))) (/ (/ l (sin k)) (tan k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 8.2e-7) {
		tmp = ((l / t_m) / (t_m * k)) / (k * (t_m / l));
	} else {
		tmp = ((2.0 * l) / (t_m * (k * k))) * ((l / sin(k)) / 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 (k <= 8.2d-7) then
        tmp = ((l / t_m) / (t_m * k)) / (k * (t_m / l))
    else
        tmp = ((2.0d0 * l) / (t_m * (k * k))) * ((l / sin(k)) / 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 (k <= 8.2e-7) {
		tmp = ((l / t_m) / (t_m * k)) / (k * (t_m / l));
	} else {
		tmp = ((2.0 * l) / (t_m * (k * k))) * ((l / Math.sin(k)) / 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 k <= 8.2e-7:
		tmp = ((l / t_m) / (t_m * k)) / (k * (t_m / l))
	else:
		tmp = ((2.0 * l) / (t_m * (k * k))) * ((l / math.sin(k)) / 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 (k <= 8.2e-7)
		tmp = Float64(Float64(Float64(l / t_m) / Float64(t_m * k)) / Float64(k * Float64(t_m / l)));
	else
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(t_m * Float64(k * k))) * Float64(Float64(l / sin(k)) / 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 (k <= 8.2e-7)
		tmp = ((l / t_m) / (t_m * k)) / (k * (t_m / l));
	else
		tmp = ((2.0 * l) / (t_m * (k * k))) * ((l / sin(k)) / 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[k, 8.2e-7], N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(k * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.1999999999999998e-7

   1. Initial program 60.0%

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

   3. Taylor expanded in k around 0

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

      1. /-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. *-commutative N/A

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

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

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 56.5%

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

   5. Simplified 56.5%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-lowering-*.f64 69.4%

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

   7. Applied egg-rr 69.4%

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

      1. associate-/r* N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 81.1%

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

   9. Applied egg-rr 81.1%

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

       1. clear-num N/A

          \[\leadsto \frac{1}{\frac{t}{\ell}} \cdot \frac{\color{blue}{\frac{\frac{\ell}{t}}{t \cdot k}}}{k} \]

       2. frac-times N/A

          \[\leadsto \frac{1 \cdot \frac{\frac{\ell}{t}}{t \cdot k}}{\color{blue}{\frac{t}{\ell} \cdot k}} \]

       3. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

       9. /-lowering-/.f64 82.5%

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

   11. Applied egg-rr 82.5%

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

   if 8.1999999999999998e-7 < k

   1. Initial program 34.6%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. cube-unmult N/A

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

      4. associate-/r* N/A

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

      5. associate-/r* N/A

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

      6. associate-/r/ N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

   4. Applied egg-rr 42.9%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 70.7%

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

   7. Simplified 70.7%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t}}{t \cdot k}}{k \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.1% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.7 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot k}}{k \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot -0.6666666666666666}{t\_m \cdot k} \cdot \frac{\ell}{k}\\ \end{array} \end{array} \]

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 (<= k 5.7e+102)
    (/ (/ (/ l t_m) (* t_m k)) (* k (/ t_m l)))
    (* (/ (* l -0.6666666666666666) (* t_m k)) (/ 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 (k <= 5.7e+102) {
		tmp = ((l / t_m) / (t_m * k)) / (k * (t_m / l));
	} else {
		tmp = ((l * -0.6666666666666666) / (t_m * k)) * (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 (k <= 5.7d+102) then
        tmp = ((l / t_m) / (t_m * k)) / (k * (t_m / l))
    else
        tmp = ((l * (-0.6666666666666666d0)) / (t_m * k)) * (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 (k <= 5.7e+102) {
		tmp = ((l / t_m) / (t_m * k)) / (k * (t_m / l));
	} else {
		tmp = ((l * -0.6666666666666666) / (t_m * k)) * (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 k <= 5.7e+102:
		tmp = ((l / t_m) / (t_m * k)) / (k * (t_m / l))
	else:
		tmp = ((l * -0.6666666666666666) / (t_m * k)) * (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 (k <= 5.7e+102)
		tmp = Float64(Float64(Float64(l / t_m) / Float64(t_m * k)) / Float64(k * Float64(t_m / l)));
	else
		tmp = Float64(Float64(Float64(l * -0.6666666666666666) / Float64(t_m * k)) * 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 (k <= 5.7e+102)
		tmp = ((l / t_m) / (t_m * k)) / (k * (t_m / l));
	else
		tmp = ((l * -0.6666666666666666) / (t_m * k)) * (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[k, 5.7e+102], N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(k * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * -0.6666666666666666), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.6999999999999999e102

   1. Initial program 58.8%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. *-commutative N/A

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

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

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 56.1%

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

   5. Simplified 56.1%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-lowering-*.f64 68.0%

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

   7. Applied egg-rr 68.0%

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

      1. associate-/r* N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 78.8%

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

   9. Applied egg-rr 78.8%

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

       1. clear-num N/A

          \[\leadsto \frac{1}{\frac{t}{\ell}} \cdot \frac{\color{blue}{\frac{\frac{\ell}{t}}{t \cdot k}}}{k} \]

       2. frac-times N/A

          \[\leadsto \frac{1 \cdot \frac{\frac{\ell}{t}}{t \cdot k}}{\color{blue}{\frac{t}{\ell} \cdot k}} \]

       3. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

       9. /-lowering-/.f64 80.1%

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

   11. Applied egg-rr 80.1%

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

   if 5.6999999999999999e102 < k

   1. Initial program 31.3%

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

      1. associate-/r* N/A

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

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

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

      3. associate-/r* N/A

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

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

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

      5. associate-/r* N/A

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

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

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

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

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

      8. associate-/r* N/A

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

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

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

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

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

      11. cube-mult N/A

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

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

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

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

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

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

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

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

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

   3. Simplified 36.1%

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

   5. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. *-commutative N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 37.5%

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

   7. Simplified 37.5%

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

   8. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

   10. Simplified 15.4%

       \[\leadsto \frac{\color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)} + \left(k \cdot k\right) \cdot \frac{-0.6666666666666666 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)}}{k \cdot k}}}{2 + k \cdot \frac{k}{t \cdot t}} \]

   11. Taylor expanded in k around inf

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

       1. associate-*r/ N/A

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

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

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

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

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

       4. unpow2 N/A

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

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

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 55.6%

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

   13. Simplified 55.6%

       \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot k\right)}} \]
   14. Step-by-step derivation

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. times-frac N/A

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

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

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

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

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

       6. *-commutative N/A

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

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

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

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

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

       9. /-lowering-/.f64 61.3%

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

   15. Applied egg-rr 61.3%

       \[\leadsto \color{blue}{\frac{\ell \cdot -0.6666666666666666}{t \cdot k} \cdot \frac{\ell}{k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.7 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t}}{t \cdot k}}{k \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot -0.6666666666666666}{t \cdot k} \cdot \frac{\ell}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.3% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7.9 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{t\_m \cdot k} \cdot \frac{\frac{\ell}{t\_m}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot -0.6666666666666666}{t\_m \cdot k} \cdot \frac{\ell}{k}\\ \end{array} \end{array} \]

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 (<= k 7.9e+99)
    (* (/ (/ l t_m) (* t_m k)) (/ (/ l t_m) k))
    (* (/ (* l -0.6666666666666666) (* t_m k)) (/ 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 (k <= 7.9e+99) {
		tmp = ((l / t_m) / (t_m * k)) * ((l / t_m) / k);
	} else {
		tmp = ((l * -0.6666666666666666) / (t_m * k)) * (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 (k <= 7.9d+99) then
        tmp = ((l / t_m) / (t_m * k)) * ((l / t_m) / k)
    else
        tmp = ((l * (-0.6666666666666666d0)) / (t_m * k)) * (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 (k <= 7.9e+99) {
		tmp = ((l / t_m) / (t_m * k)) * ((l / t_m) / k);
	} else {
		tmp = ((l * -0.6666666666666666) / (t_m * k)) * (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 k <= 7.9e+99:
		tmp = ((l / t_m) / (t_m * k)) * ((l / t_m) / k)
	else:
		tmp = ((l * -0.6666666666666666) / (t_m * k)) * (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 (k <= 7.9e+99)
		tmp = Float64(Float64(Float64(l / t_m) / Float64(t_m * k)) * Float64(Float64(l / t_m) / k));
	else
		tmp = Float64(Float64(Float64(l * -0.6666666666666666) / Float64(t_m * k)) * 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 (k <= 7.9e+99)
		tmp = ((l / t_m) / (t_m * k)) * ((l / t_m) / k);
	else
		tmp = ((l * -0.6666666666666666) / (t_m * k)) * (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[k, 7.9e+99], N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * -0.6666666666666666), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.9000000000000003e99

   1. Initial program 58.8%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. *-commutative N/A

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

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

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 56.1%

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

   5. Simplified 56.1%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-lowering-*.f64 68.0%

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

   7. Applied egg-rr 68.0%

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

      1. associate-/r* N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 78.8%

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

   9. Applied egg-rr 78.8%

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

       1. *-commutative N/A

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

       2. clear-num N/A

          \[\leadsto \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{k} \cdot \frac{1}{\color{blue}{\frac{t}{\ell}}} \]

       3. frac-times N/A

          \[\leadsto \frac{\frac{\frac{\ell}{t}}{t \cdot k} \cdot 1}{\color{blue}{k \cdot \frac{t}{\ell}}} \]

       4. div-inv N/A

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

       5. *-rgt-identity N/A

          \[\leadsto \frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{\color{blue}{1}}{k \cdot \frac{t}{\ell}} \]

       6. clear-num N/A

          \[\leadsto \frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{1}{k \cdot \frac{1}{\color{blue}{\frac{\ell}{t}}}} \]

       7. div-inv N/A

          \[\leadsto \frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{1}{\frac{k}{\color{blue}{\frac{\ell}{t}}}} \]

       8. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

       14. /-lowering-/.f64 80.1%

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

   11. Applied egg-rr 80.1%

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

   if 7.9000000000000003e99 < k

   1. Initial program 31.3%

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

      1. associate-/r* N/A

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

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

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

      3. associate-/r* N/A

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

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

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

      5. associate-/r* N/A

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

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

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

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

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

      8. associate-/r* N/A

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

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

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

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

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

      11. cube-mult N/A

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

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

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

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

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

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

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

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

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

   3. Simplified 36.1%

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

   5. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. *-commutative N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 37.5%

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

   7. Simplified 37.5%

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

   8. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

   10. Simplified 15.4%

       \[\leadsto \frac{\color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)} + \left(k \cdot k\right) \cdot \frac{-0.6666666666666666 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)}}{k \cdot k}}}{2 + k \cdot \frac{k}{t \cdot t}} \]

   11. Taylor expanded in k around inf

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

       1. associate-*r/ N/A

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

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

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

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

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

       4. unpow2 N/A

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

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

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 55.6%

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

   13. Simplified 55.6%

       \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot k\right)}} \]
   14. Step-by-step derivation

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. times-frac N/A

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

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

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

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

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

       6. *-commutative N/A

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

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

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

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

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

       9. /-lowering-/.f64 61.3%

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

   15. Applied egg-rr 61.3%

       \[\leadsto \color{blue}{\frac{\ell \cdot -0.6666666666666666}{t \cdot k} \cdot \frac{\ell}{k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 72.3% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.7 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\frac{\frac{\ell}{t\_m \cdot k}}{t\_m}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot -0.6666666666666666}{t\_m \cdot k} \cdot \frac{\ell}{k}\\ \end{array} \end{array} \]

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 (<= k 5.7e+102)
    (* (/ l t_m) (/ (/ (/ l (* t_m k)) t_m) k))
    (* (/ (* l -0.6666666666666666) (* t_m k)) (/ 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 (k <= 5.7e+102) {
		tmp = (l / t_m) * (((l / (t_m * k)) / t_m) / k);
	} else {
		tmp = ((l * -0.6666666666666666) / (t_m * k)) * (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 (k <= 5.7d+102) then
        tmp = (l / t_m) * (((l / (t_m * k)) / t_m) / k)
    else
        tmp = ((l * (-0.6666666666666666d0)) / (t_m * k)) * (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 (k <= 5.7e+102) {
		tmp = (l / t_m) * (((l / (t_m * k)) / t_m) / k);
	} else {
		tmp = ((l * -0.6666666666666666) / (t_m * k)) * (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 k <= 5.7e+102:
		tmp = (l / t_m) * (((l / (t_m * k)) / t_m) / k)
	else:
		tmp = ((l * -0.6666666666666666) / (t_m * k)) * (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 (k <= 5.7e+102)
		tmp = Float64(Float64(l / t_m) * Float64(Float64(Float64(l / Float64(t_m * k)) / t_m) / k));
	else
		tmp = Float64(Float64(Float64(l * -0.6666666666666666) / Float64(t_m * k)) * 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 (k <= 5.7e+102)
		tmp = (l / t_m) * (((l / (t_m * k)) / t_m) / k);
	else
		tmp = ((l * -0.6666666666666666) / (t_m * k)) * (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[k, 5.7e+102], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * -0.6666666666666666), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.6999999999999999e102

   1. Initial program 58.8%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. *-commutative N/A

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

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

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 56.1%

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

   5. Simplified 56.1%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-lowering-*.f64 68.0%

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

   7. Applied egg-rr 68.0%

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

      1. associate-/r* N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 78.8%

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

   9. Applied egg-rr 78.8%

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

       1. associate-/l/ N/A

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

       2. associate-/r* N/A

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

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

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

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

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

       5. *-lowering-*.f64 78.9%

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

   11. Applied egg-rr 78.9%

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

   if 5.6999999999999999e102 < k

   1. Initial program 31.3%

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

      1. associate-/r* N/A

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

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

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

      3. associate-/r* N/A

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

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

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

      5. associate-/r* N/A

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

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

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

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

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

      8. associate-/r* N/A

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

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

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

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

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

      11. cube-mult N/A

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

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

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

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

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

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

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

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

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

   3. Simplified 36.1%

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

   5. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. *-commutative N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 37.5%

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

   7. Simplified 37.5%

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

   8. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

   10. Simplified 15.4%

       \[\leadsto \frac{\color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)} + \left(k \cdot k\right) \cdot \frac{-0.6666666666666666 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)}}{k \cdot k}}}{2 + k \cdot \frac{k}{t \cdot t}} \]

   11. Taylor expanded in k around inf

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

       1. associate-*r/ N/A

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

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

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

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

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

       4. unpow2 N/A

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

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

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 55.6%

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

   13. Simplified 55.6%

       \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot k\right)}} \]
   14. Step-by-step derivation

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. times-frac N/A

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

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

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

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

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

       6. *-commutative N/A

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

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

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

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

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

       9. /-lowering-/.f64 61.3%

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

   15. Applied egg-rr 61.3%

       \[\leadsto \color{blue}{\frac{\ell \cdot -0.6666666666666666}{t \cdot k} \cdot \frac{\ell}{k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 72.3% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{+104}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot -0.6666666666666666}{t\_m \cdot k} \cdot \frac{\ell}{k}\\ \end{array} \end{array} \]

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 (<= k 2.7e+104)
    (* (/ l t_m) (/ (/ (/ l t_m) (* t_m k)) k))
    (* (/ (* l -0.6666666666666666) (* t_m k)) (/ 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 (k <= 2.7e+104) {
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k);
	} else {
		tmp = ((l * -0.6666666666666666) / (t_m * k)) * (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 (k <= 2.7d+104) then
        tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k)
    else
        tmp = ((l * (-0.6666666666666666d0)) / (t_m * k)) * (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 (k <= 2.7e+104) {
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k);
	} else {
		tmp = ((l * -0.6666666666666666) / (t_m * k)) * (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 k <= 2.7e+104:
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k)
	else:
		tmp = ((l * -0.6666666666666666) / (t_m * k)) * (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 (k <= 2.7e+104)
		tmp = Float64(Float64(l / t_m) * Float64(Float64(Float64(l / t_m) / Float64(t_m * k)) / k));
	else
		tmp = Float64(Float64(Float64(l * -0.6666666666666666) / Float64(t_m * k)) * 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 (k <= 2.7e+104)
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k);
	else
		tmp = ((l * -0.6666666666666666) / (t_m * k)) * (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[k, 2.7e+104], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * -0.6666666666666666), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.69999999999999985e104

   1. Initial program 58.8%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. *-commutative N/A

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

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

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 56.1%

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

   5. Simplified 56.1%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-lowering-*.f64 68.0%

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

   7. Applied egg-rr 68.0%

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

      1. associate-/r* N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 78.8%

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

   9. Applied egg-rr 78.8%

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

   if 2.69999999999999985e104 < k

   1. Initial program 31.3%

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

      1. associate-/r* N/A

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

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

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

      3. associate-/r* N/A

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

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

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

      5. associate-/r* N/A

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

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

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

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

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

      8. associate-/r* N/A

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

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

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

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

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

      11. cube-mult N/A

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

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

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

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

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

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

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

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

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

   3. Simplified 36.1%

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

   5. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. *-commutative N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 37.5%

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

   7. Simplified 37.5%

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

   8. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

   10. Simplified 15.4%

       \[\leadsto \frac{\color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)} + \left(k \cdot k\right) \cdot \frac{-0.6666666666666666 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)}}{k \cdot k}}}{2 + k \cdot \frac{k}{t \cdot t}} \]

   11. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{-2}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2}{3} \cdot {\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \left({\ell}^{2}\right)\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \left(\ell \cdot \ell\right)\right), \left({k}^{\color{blue}{2}} \cdot t\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({k}^{\color{blue}{2}} \cdot t\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(t \cdot \color{blue}{{k}^{2}}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \color{blue}{\left({k}^{2}\right)}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \left(k \cdot \color{blue}{k}\right)\right)\right) \]

       9. *-lowering-*.f64 55.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   13. Simplified 55.6%

       \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot k\right)}} \]
   14. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\left(\frac{-2}{3} \cdot \ell\right) \cdot \ell}{\color{blue}{t} \cdot \left(k \cdot k\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{\left(\frac{-2}{3} \cdot \ell\right) \cdot \ell}{\left(t \cdot k\right) \cdot \color{blue}{k}} \]

       3. times-frac N/A

          \[\leadsto \frac{\frac{-2}{3} \cdot \ell}{t \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-2}{3} \cdot \ell}{t \cdot k}\right), \color{blue}{\left(\frac{\ell}{k}\right)}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{3} \cdot \ell\right), \left(t \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{k}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \frac{-2}{3}\right), \left(t \cdot k\right)\right), \left(\frac{\ell}{k}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \frac{-2}{3}\right), \left(t \cdot k\right)\right), \left(\frac{\ell}{k}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \frac{-2}{3}\right), \mathsf{*.f64}\left(t, k\right)\right), \left(\frac{\ell}{k}\right)\right) \]

       9. /-lowering-/.f64 61.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \frac{-2}{3}\right), \mathsf{*.f64}\left(t, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{k}\right)\right) \]

   15. Applied egg-rr 61.3%

       \[\leadsto \color{blue}{\frac{\ell \cdot -0.6666666666666666}{t \cdot k} \cdot \frac{\ell}{k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 70.1% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.3 \cdot 10^{+99}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\frac{\ell}{t\_m \cdot \left(t\_m \cdot k\right)}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot -0.6666666666666666}{t\_m \cdot k} \cdot \frac{\ell}{k}\\ \end{array} \end{array} \]

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 (<= k 5.3e+99)
    (* (/ l t_m) (/ (/ l (* t_m (* t_m k))) k))
    (* (/ (* l -0.6666666666666666) (* t_m k)) (/ 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 (k <= 5.3e+99) {
		tmp = (l / t_m) * ((l / (t_m * (t_m * k))) / k);
	} else {
		tmp = ((l * -0.6666666666666666) / (t_m * k)) * (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 (k <= 5.3d+99) then
        tmp = (l / t_m) * ((l / (t_m * (t_m * k))) / k)
    else
        tmp = ((l * (-0.6666666666666666d0)) / (t_m * k)) * (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 (k <= 5.3e+99) {
		tmp = (l / t_m) * ((l / (t_m * (t_m * k))) / k);
	} else {
		tmp = ((l * -0.6666666666666666) / (t_m * k)) * (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 k <= 5.3e+99:
		tmp = (l / t_m) * ((l / (t_m * (t_m * k))) / k)
	else:
		tmp = ((l * -0.6666666666666666) / (t_m * k)) * (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 (k <= 5.3e+99)
		tmp = Float64(Float64(l / t_m) * Float64(Float64(l / Float64(t_m * Float64(t_m * k))) / k));
	else
		tmp = Float64(Float64(Float64(l * -0.6666666666666666) / Float64(t_m * k)) * 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 (k <= 5.3e+99)
		tmp = (l / t_m) * ((l / (t_m * (t_m * k))) / k);
	else
		tmp = ((l * -0.6666666666666666) / (t_m * k)) * (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[k, 5.3e+99], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(l / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * -0.6666666666666666), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.30000000000000034e99

   1. Initial program 58.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      12. *-lowering-*.f64 56.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   5. Simplified 56.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 68.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 68.0%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\frac{\ell}{t}}{\color{blue}{t \cdot \left(k \cdot k\right)}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\frac{\ell}{t}}{\left(t \cdot k\right) \cdot \color{blue}{k}}\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\frac{\frac{\ell}{t}}{t \cdot k}}{\color{blue}{k}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{t \cdot k}\right), \color{blue}{k}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(t \cdot k\right)\right), k\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(t \cdot k\right)\right), k\right)\right) \]

      7. *-lowering-*.f64 78.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, k\right)\right), k\right)\right) \]

   9. Applied egg-rr 78.8%

      \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\frac{\frac{\ell}{t}}{t \cdot k}}{k}} \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot \left(t \cdot k\right)}\right), k\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot k\right)\right)\right), k\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot k\right)\right)\right), k\right)\right) \]

       4. *-lowering-*.f64 76.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, k\right)\right)\right), k\right)\right) \]

   11. Applied egg-rr 76.9%

       \[\leadsto \frac{\ell}{t} \cdot \frac{\color{blue}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}{k} \]

   if 5.30000000000000034e99 < k

   1. Initial program 31.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}\right), \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k}\right), \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right), \tan k\right), \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\sin k}\right), \tan k\right), \left(\left(\color{blue}{1} + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}\right), \sin k\right), \tan k\right), \left(\left(\color{blue}{1} + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{{t}^{3}}{\ell \cdot \ell}\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{\frac{{t}^{3}}{\ell}}{\ell}\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{t}^{3}}{\ell}\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({t}^{3}\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      11. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      14. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \ell\right)\right), \mathsf{sin.f64}\left(k\right)\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      15. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \ell\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \left(\left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right) + 1\right)\right) \]

   3. Simplified 36.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{2}{\frac{\frac{t \cdot \left(t \cdot t\right)}{\ell}}{\ell}}}{\sin k}}{\tan k}}{2 + k \cdot \frac{k}{t \cdot t}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{k \cdot {t}^{3}}\right)}, \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot {\ell}^{2}}{k \cdot {t}^{3}}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot {\ell}^{2}\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({t}^{3} \cdot k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\left(t \cdot {t}^{2}\right) \cdot k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(t \cdot \left({t}^{2} \cdot k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \left({t}^{2} \cdot k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left({t}^{2}\right), k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot t\right), k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 37.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   7. Simplified 37.5%

      \[\leadsto \frac{\frac{\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(t \cdot t\right) \cdot k\right)}}}{\tan k}}{2 + k \cdot \frac{k}{t \cdot t}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-2}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right)}, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + \frac{-2}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} \cdot \frac{-2}{3}}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + \left({k}^{2} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) \cdot \frac{-2}{3}}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + {k}^{2} \cdot \left(\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{-2}{3}\right)}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + {k}^{2} \cdot \left(\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right)}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + {k}^{2} \cdot \left(\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right)\right), \left({k}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   10. Simplified 15.4%

       \[\leadsto \frac{\color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)} + \left(k \cdot k\right) \cdot \frac{-0.6666666666666666 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)}}{k \cdot k}}}{2 + k \cdot \frac{k}{t \cdot t}} \]

   11. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{-2}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2}{3} \cdot {\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \left({\ell}^{2}\right)\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \left(\ell \cdot \ell\right)\right), \left({k}^{\color{blue}{2}} \cdot t\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({k}^{\color{blue}{2}} \cdot t\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(t \cdot \color{blue}{{k}^{2}}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \color{blue}{\left({k}^{2}\right)}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \left(k \cdot \color{blue}{k}\right)\right)\right) \]

       9. *-lowering-*.f64 55.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   13. Simplified 55.6%

       \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot k\right)}} \]
   14. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\left(\frac{-2}{3} \cdot \ell\right) \cdot \ell}{\color{blue}{t} \cdot \left(k \cdot k\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{\left(\frac{-2}{3} \cdot \ell\right) \cdot \ell}{\left(t \cdot k\right) \cdot \color{blue}{k}} \]

       3. times-frac N/A

          \[\leadsto \frac{\frac{-2}{3} \cdot \ell}{t \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-2}{3} \cdot \ell}{t \cdot k}\right), \color{blue}{\left(\frac{\ell}{k}\right)}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{3} \cdot \ell\right), \left(t \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{k}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \frac{-2}{3}\right), \left(t \cdot k\right)\right), \left(\frac{\ell}{k}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \frac{-2}{3}\right), \left(t \cdot k\right)\right), \left(\frac{\ell}{k}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \frac{-2}{3}\right), \mathsf{*.f64}\left(t, k\right)\right), \left(\frac{\ell}{k}\right)\right) \]

       9. /-lowering-/.f64 61.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \frac{-2}{3}\right), \mathsf{*.f64}\left(t, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{k}\right)\right) \]

   15. Applied egg-rr 61.3%

       \[\leadsto \color{blue}{\frac{\ell \cdot -0.6666666666666666}{t \cdot k} \cdot \frac{\ell}{k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 70.1% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{+104}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot -0.6666666666666666}{t\_m \cdot k} \cdot \frac{\ell}{k}\\ \end{array} \end{array} \]

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 (<= k 3.5e+104)
    (* (/ l t_m) (/ l (* (* t_m k) (* t_m k))))
    (* (/ (* l -0.6666666666666666) (* t_m k)) (/ 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 (k <= 3.5e+104) {
		tmp = (l / t_m) * (l / ((t_m * k) * (t_m * k)));
	} else {
		tmp = ((l * -0.6666666666666666) / (t_m * k)) * (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 (k <= 3.5d+104) then
        tmp = (l / t_m) * (l / ((t_m * k) * (t_m * k)))
    else
        tmp = ((l * (-0.6666666666666666d0)) / (t_m * k)) * (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 (k <= 3.5e+104) {
		tmp = (l / t_m) * (l / ((t_m * k) * (t_m * k)));
	} else {
		tmp = ((l * -0.6666666666666666) / (t_m * k)) * (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 k <= 3.5e+104:
		tmp = (l / t_m) * (l / ((t_m * k) * (t_m * k)))
	else:
		tmp = ((l * -0.6666666666666666) / (t_m * k)) * (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 (k <= 3.5e+104)
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(Float64(t_m * k) * Float64(t_m * k))));
	else
		tmp = Float64(Float64(Float64(l * -0.6666666666666666) / Float64(t_m * k)) * 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 (k <= 3.5e+104)
		tmp = (l / t_m) * (l / ((t_m * k) * (t_m * k)));
	else
		tmp = ((l * -0.6666666666666666) / (t_m * k)) * (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[k, 3.5e+104], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * -0.6666666666666666), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.5000000000000002e104

   1. Initial program 58.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      12. *-lowering-*.f64 56.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   5. Simplified 56.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 68.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 68.0%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)\right) \]

      2. unswap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \left(\left(t \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot k\right), \color{blue}{\left(t \cdot k\right)}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \left(\color{blue}{t} \cdot k\right)\right)\right)\right) \]

      5. *-lowering-*.f64 74.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \mathsf{*.f64}\left(t, \color{blue}{k}\right)\right)\right)\right) \]

   9. Applied egg-rr 74.3%

      \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}} \]

   if 3.5000000000000002e104 < k

   1. Initial program 31.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}\right), \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k}\right), \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right), \tan k\right), \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\sin k}\right), \tan k\right), \left(\left(\color{blue}{1} + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}\right), \sin k\right), \tan k\right), \left(\left(\color{blue}{1} + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{{t}^{3}}{\ell \cdot \ell}\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{\frac{{t}^{3}}{\ell}}{\ell}\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{t}^{3}}{\ell}\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({t}^{3}\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      11. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      14. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \ell\right)\right), \mathsf{sin.f64}\left(k\right)\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      15. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \ell\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \left(\left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right) + 1\right)\right) \]

   3. Simplified 36.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{2}{\frac{\frac{t \cdot \left(t \cdot t\right)}{\ell}}{\ell}}}{\sin k}}{\tan k}}{2 + k \cdot \frac{k}{t \cdot t}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{k \cdot {t}^{3}}\right)}, \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot {\ell}^{2}}{k \cdot {t}^{3}}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot {\ell}^{2}\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({t}^{3} \cdot k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\left(t \cdot {t}^{2}\right) \cdot k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(t \cdot \left({t}^{2} \cdot k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \left({t}^{2} \cdot k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left({t}^{2}\right), k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot t\right), k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 37.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   7. Simplified 37.5%

      \[\leadsto \frac{\frac{\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(t \cdot t\right) \cdot k\right)}}}{\tan k}}{2 + k \cdot \frac{k}{t \cdot t}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-2}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right)}, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + \frac{-2}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} \cdot \frac{-2}{3}}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + \left({k}^{2} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) \cdot \frac{-2}{3}}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + {k}^{2} \cdot \left(\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{-2}{3}\right)}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + {k}^{2} \cdot \left(\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right)}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + {k}^{2} \cdot \left(\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right)\right), \left({k}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   10. Simplified 15.4%

       \[\leadsto \frac{\color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)} + \left(k \cdot k\right) \cdot \frac{-0.6666666666666666 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)}}{k \cdot k}}}{2 + k \cdot \frac{k}{t \cdot t}} \]

   11. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{-2}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2}{3} \cdot {\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \left({\ell}^{2}\right)\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \left(\ell \cdot \ell\right)\right), \left({k}^{\color{blue}{2}} \cdot t\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({k}^{\color{blue}{2}} \cdot t\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(t \cdot \color{blue}{{k}^{2}}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \color{blue}{\left({k}^{2}\right)}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \left(k \cdot \color{blue}{k}\right)\right)\right) \]

       9. *-lowering-*.f64 55.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   13. Simplified 55.6%

       \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot k\right)}} \]
   14. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\left(\frac{-2}{3} \cdot \ell\right) \cdot \ell}{\color{blue}{t} \cdot \left(k \cdot k\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{\left(\frac{-2}{3} \cdot \ell\right) \cdot \ell}{\left(t \cdot k\right) \cdot \color{blue}{k}} \]

       3. times-frac N/A

          \[\leadsto \frac{\frac{-2}{3} \cdot \ell}{t \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-2}{3} \cdot \ell}{t \cdot k}\right), \color{blue}{\left(\frac{\ell}{k}\right)}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{3} \cdot \ell\right), \left(t \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{k}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \frac{-2}{3}\right), \left(t \cdot k\right)\right), \left(\frac{\ell}{k}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \frac{-2}{3}\right), \left(t \cdot k\right)\right), \left(\frac{\ell}{k}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \frac{-2}{3}\right), \mathsf{*.f64}\left(t, k\right)\right), \left(\frac{\ell}{k}\right)\right) \]

       9. /-lowering-/.f64 61.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \frac{-2}{3}\right), \mathsf{*.f64}\left(t, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{k}\right)\right) \]

   15. Applied egg-rr 61.3%

       \[\leadsto \color{blue}{\frac{\ell \cdot -0.6666666666666666}{t \cdot k} \cdot \frac{\ell}{k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 50.0% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 4.3 \cdot 10^{-215}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \left(\ell \cdot -0.6666666666666666\right)}{t\_m}}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 4.3e-215)
    (/ (/ (/ (* l (* l -0.6666666666666666)) t_m) k) k)
    (* (/ l t_m) (/ l (* 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 (l <= 4.3e-215) {
		tmp = (((l * (l * -0.6666666666666666)) / t_m) / k) / k;
	} else {
		tmp = (l / t_m) * (l / (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 (l <= 4.3d-215) then
        tmp = (((l * (l * (-0.6666666666666666d0))) / t_m) / k) / k
    else
        tmp = (l / t_m) * (l / (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 (l <= 4.3e-215) {
		tmp = (((l * (l * -0.6666666666666666)) / t_m) / k) / k;
	} else {
		tmp = (l / t_m) * (l / (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 l <= 4.3e-215:
		tmp = (((l * (l * -0.6666666666666666)) / t_m) / k) / k
	else:
		tmp = (l / t_m) * (l / (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 (l <= 4.3e-215)
		tmp = Float64(Float64(Float64(Float64(l * Float64(l * -0.6666666666666666)) / t_m) / k) / k);
	else
		tmp = Float64(Float64(l / t_m) * Float64(l / 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 (l <= 4.3e-215)
		tmp = (((l * (l * -0.6666666666666666)) / t_m) / k) / k;
	else
		tmp = (l / t_m) * (l / (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[l, 4.3e-215], N[(N[(N[(N[(l * N[(l * -0.6666666666666666), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4.30000000000000024e-215

   1. Initial program 58.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}\right), \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k}\right), \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right), \tan k\right), \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\sin k}\right), \tan k\right), \left(\left(\color{blue}{1} + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}\right), \sin k\right), \tan k\right), \left(\left(\color{blue}{1} + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{{t}^{3}}{\ell \cdot \ell}\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{\frac{{t}^{3}}{\ell}}{\ell}\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{t}^{3}}{\ell}\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({t}^{3}\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      11. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      14. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \ell\right)\right), \mathsf{sin.f64}\left(k\right)\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

      15. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \ell\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \left(\left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right) + 1\right)\right) \]

   3. Simplified 56.0%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{2}{\frac{\frac{t \cdot \left(t \cdot t\right)}{\ell}}{\ell}}}{\sin k}}{\tan k}}{2 + k \cdot \frac{k}{t \cdot t}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{k \cdot {t}^{3}}\right)}, \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot {\ell}^{2}}{k \cdot {t}^{3}}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot {\ell}^{2}\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({t}^{3} \cdot k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\left(t \cdot {t}^{2}\right) \cdot k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(t \cdot \left({t}^{2} \cdot k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \left({t}^{2} \cdot k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left({t}^{2}\right), k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot t\right), k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 50.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   7. Simplified 50.4%

      \[\leadsto \frac{\frac{\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(t \cdot t\right) \cdot k\right)}}}{\tan k}}{2 + k \cdot \frac{k}{t \cdot t}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-2}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right)}, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + \frac{-2}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} \cdot \frac{-2}{3}}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + \left({k}^{2} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) \cdot \frac{-2}{3}}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + {k}^{2} \cdot \left(\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{-2}{3}\right)}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + {k}^{2} \cdot \left(\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right)}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + {k}^{2} \cdot \left(\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right)\right), \left({k}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   10. Simplified 26.6%

       \[\leadsto \frac{\color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)} + \left(k \cdot k\right) \cdot \frac{-0.6666666666666666 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)}}{k \cdot k}}}{2 + k \cdot \frac{k}{t \cdot t}} \]

   11. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{-2}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2}{3} \cdot {\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \left({\ell}^{2}\right)\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \left(\ell \cdot \ell\right)\right), \left({k}^{\color{blue}{2}} \cdot t\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({k}^{\color{blue}{2}} \cdot t\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(t \cdot \color{blue}{{k}^{2}}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \color{blue}{\left({k}^{2}\right)}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \left(k \cdot \color{blue}{k}\right)\right)\right) \]

       9. *-lowering-*.f64 35.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   13. Simplified 35.4%

       \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot k\right)}} \]
   14. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{-2}{3} \cdot \left(\ell \cdot \ell\right)}{t}}{\color{blue}{k \cdot k}} \]

       2. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\frac{-2}{3} \cdot \left(\ell \cdot \ell\right)}{t}}{k}}{\color{blue}{k}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{-2}{3} \cdot \left(\ell \cdot \ell\right)}{t}}{k}\right), \color{blue}{k}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-2}{3} \cdot \left(\ell \cdot \ell\right)}{t}\right), k\right), k\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{3} \cdot \left(\ell \cdot \ell\right)\right), t\right), k\right), k\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{-2}{3} \cdot \ell\right) \cdot \ell\right), t\right), k\right), k\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \left(\frac{-2}{3} \cdot \ell\right)\right), t\right), k\right), k\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{-2}{3} \cdot \ell\right)\right), t\right), k\right), k\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{-2}{3}\right)\right), t\right), k\right), k\right) \]

       10. *-lowering-*.f64 40.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{-2}{3}\right)\right), t\right), k\right), k\right) \]

   15. Applied egg-rr 40.9%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\ell \cdot \left(\ell \cdot -0.6666666666666666\right)}{t}}{k}}{k}} \]

   if 4.30000000000000024e-215 < l

   1. Initial program 47.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-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. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      12. *-lowering-*.f64 49.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   5. Simplified 49.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 61.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 61.8%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 35.3% accurate, 38.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\frac{\frac{\ell \cdot \left(\ell \cdot -0.6666666666666666\right)}{t\_m}}{k}}{k} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (/ (/ (* l (* l -0.6666666666666666)) 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) {
	return t_s * ((((l * (l * -0.6666666666666666)) / t_m) / k) / k);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((((l * (l * (-0.6666666666666666d0))) / t_m) / k) / k)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((((l * (l * -0.6666666666666666)) / t_m) / k) / k);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((((l * (l * -0.6666666666666666)) / t_m) / k) / k)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(Float64(Float64(l * Float64(l * -0.6666666666666666)) / t_m) / k) / k))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((((l * (l * -0.6666666666666666)) / t_m) / k) / k);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(N[(N[(l * N[(l * -0.6666666666666666), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}\right), \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k}\right), \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right), \tan k\right), \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right) \]

   5. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\sin k}\right), \tan k\right), \left(\left(\color{blue}{1} + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}\right), \sin k\right), \tan k\right), \left(\left(\color{blue}{1} + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{{t}^{3}}{\ell \cdot \ell}\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   8. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{\frac{{t}^{3}}{\ell}}{\ell}\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{t}^{3}}{\ell}\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({t}^{3}\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   11. cube-mult N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   14. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \ell\right)\right), \mathsf{sin.f64}\left(k\right)\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   15. tan-lowering-tan.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \ell\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \left(\left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right) + 1\right)\right) \]

3. Simplified 51.0%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{2}{\frac{\frac{t \cdot \left(t \cdot t\right)}{\ell}}{\ell}}}{\sin k}}{\tan k}}{2 + k \cdot \frac{k}{t \cdot t}}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{k \cdot {t}^{3}}\right)}, \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot {\ell}^{2}}{k \cdot {t}^{3}}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot {\ell}^{2}\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({t}^{3} \cdot k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   7. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\left(t \cdot {t}^{2}\right) \cdot k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(t \cdot \left({t}^{2} \cdot k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \left({t}^{2} \cdot k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left({t}^{2}\right), k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot t\right), k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 47.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

7. Simplified 47.2%

   \[\leadsto \frac{\frac{\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(t \cdot t\right) \cdot k\right)}}}{\tan k}}{2 + k \cdot \frac{k}{t \cdot t}} \]

8. Taylor expanded in k around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-2}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right)}, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + \frac{-2}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} \cdot \frac{-2}{3}}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   3. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + \left({k}^{2} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) \cdot \frac{-2}{3}}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + {k}^{2} \cdot \left(\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{-2}{3}\right)}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + {k}^{2} \cdot \left(\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right)}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + {k}^{2} \cdot \left(\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right)\right), \left({k}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

10. Simplified 22.7%

    \[\leadsto \frac{\color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)} + \left(k \cdot k\right) \cdot \frac{-0.6666666666666666 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)}}{k \cdot k}}}{2 + k \cdot \frac{k}{t \cdot t}} \]

11. Taylor expanded in k around inf

    \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
12. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{\frac{-2}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

    2. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2}{3} \cdot {\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \left({\ell}^{2}\right)\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \left(\ell \cdot \ell\right)\right), \left({k}^{\color{blue}{2}} \cdot t\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({k}^{\color{blue}{2}} \cdot t\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(t \cdot \color{blue}{{k}^{2}}\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \color{blue}{\left({k}^{2}\right)}\right)\right) \]

    8. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \left(k \cdot \color{blue}{k}\right)\right)\right) \]

    9. *-lowering-*.f64 26.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

13. Simplified 26.9%

    \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot k\right)}} \]
14. Step-by-step derivation

    1. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{-2}{3} \cdot \left(\ell \cdot \ell\right)}{t}}{\color{blue}{k \cdot k}} \]

    2. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{\frac{-2}{3} \cdot \left(\ell \cdot \ell\right)}{t}}{k}}{\color{blue}{k}} \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{-2}{3} \cdot \left(\ell \cdot \ell\right)}{t}}{k}\right), \color{blue}{k}\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-2}{3} \cdot \left(\ell \cdot \ell\right)}{t}\right), k\right), k\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{3} \cdot \left(\ell \cdot \ell\right)\right), t\right), k\right), k\right) \]

    6. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{-2}{3} \cdot \ell\right) \cdot \ell\right), t\right), k\right), k\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \left(\frac{-2}{3} \cdot \ell\right)\right), t\right), k\right), k\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{-2}{3} \cdot \ell\right)\right), t\right), k\right), k\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{-2}{3}\right)\right), t\right), k\right), k\right) \]

    10. *-lowering-*.f64 31.1%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{-2}{3}\right)\right), t\right), k\right), k\right) \]

15. Applied egg-rr 31.1%

    \[\leadsto \color{blue}{\frac{\frac{\frac{\ell \cdot \left(\ell \cdot -0.6666666666666666\right)}{t}}{k}}{k}} \]
16. Add Preprocessing

Alternative 12: 34.6% accurate, 38.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\ell \cdot \ell}{t\_m \cdot k} \cdot \frac{-0.6666666666666666}{k}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (/ (* l l) (* t_m k)) (/ -0.6666666666666666 k))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (((l * l) / (t_m * k)) * (-0.6666666666666666 / k));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (((l * l) / (t_m * k)) * ((-0.6666666666666666d0) / k))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (((l * l) / (t_m * k)) * (-0.6666666666666666 / k));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (((l * l) / (t_m * k)) * (-0.6666666666666666 / k))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(Float64(l * l) / Float64(t_m * k)) * Float64(-0.6666666666666666 / k)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (((l * l) / (t_m * k)) * (-0.6666666666666666 / k));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(N[(l * l), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(-0.6666666666666666 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}\right), \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k}\right), \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right), \tan k\right), \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right) \]

   5. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\sin k}\right), \tan k\right), \left(\left(\color{blue}{1} + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}\right), \sin k\right), \tan k\right), \left(\left(\color{blue}{1} + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{{t}^{3}}{\ell \cdot \ell}\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   8. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{\frac{{t}^{3}}{\ell}}{\ell}\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{t}^{3}}{\ell}\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({t}^{3}\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   11. cube-mult N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   14. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \ell\right)\right), \mathsf{sin.f64}\left(k\right)\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   15. tan-lowering-tan.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \ell\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \left(\left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right) + 1\right)\right) \]

3. Simplified 51.0%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{2}{\frac{\frac{t \cdot \left(t \cdot t\right)}{\ell}}{\ell}}}{\sin k}}{\tan k}}{2 + k \cdot \frac{k}{t \cdot t}}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{k \cdot {t}^{3}}\right)}, \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot {\ell}^{2}}{k \cdot {t}^{3}}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot {\ell}^{2}\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({t}^{3} \cdot k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   7. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\left(t \cdot {t}^{2}\right) \cdot k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(t \cdot \left({t}^{2} \cdot k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \left({t}^{2} \cdot k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left({t}^{2}\right), k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot t\right), k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 47.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

7. Simplified 47.2%

   \[\leadsto \frac{\frac{\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(t \cdot t\right) \cdot k\right)}}}{\tan k}}{2 + k \cdot \frac{k}{t \cdot t}} \]

8. Taylor expanded in k around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-2}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right)}, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + \frac{-2}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} \cdot \frac{-2}{3}}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   3. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + \left({k}^{2} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) \cdot \frac{-2}{3}}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + {k}^{2} \cdot \left(\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{-2}{3}\right)}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + {k}^{2} \cdot \left(\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right)}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + {k}^{2} \cdot \left(\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right)\right), \left({k}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

10. Simplified 22.7%

    \[\leadsto \frac{\color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)} + \left(k \cdot k\right) \cdot \frac{-0.6666666666666666 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)}}{k \cdot k}}}{2 + k \cdot \frac{k}{t \cdot t}} \]

11. Taylor expanded in k around inf

    \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
12. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{\frac{-2}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

    2. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2}{3} \cdot {\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \left({\ell}^{2}\right)\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \left(\ell \cdot \ell\right)\right), \left({k}^{\color{blue}{2}} \cdot t\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({k}^{\color{blue}{2}} \cdot t\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(t \cdot \color{blue}{{k}^{2}}\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \color{blue}{\left({k}^{2}\right)}\right)\right) \]

    8. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \left(k \cdot \color{blue}{k}\right)\right)\right) \]

    9. *-lowering-*.f64 26.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

13. Simplified 26.9%

    \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot k\right)}} \]
14. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-2}{3}}{\color{blue}{t} \cdot \left(k \cdot k\right)} \]

    2. associate-*r* N/A

       \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-2}{3}}{\left(t \cdot k\right) \cdot \color{blue}{k}} \]

    3. times-frac N/A

       \[\leadsto \frac{\ell \cdot \ell}{t \cdot k} \cdot \color{blue}{\frac{\frac{-2}{3}}{k}} \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell \cdot \ell}{t \cdot k}\right), \color{blue}{\left(\frac{\frac{-2}{3}}{k}\right)}\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(t \cdot k\right)\right), \left(\frac{\color{blue}{\frac{-2}{3}}}{k}\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(t \cdot k\right)\right), \left(\frac{\frac{-2}{3}}{k}\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(t, k\right)\right), \left(\frac{\frac{-2}{3}}{k}\right)\right) \]

    8. /-lowering-/.f64 30.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(t, k\right)\right), \mathsf{/.f64}\left(\frac{-2}{3}, \color{blue}{k}\right)\right) \]

15. Applied egg-rr 30.7%

    \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{t \cdot k} \cdot \frac{-0.6666666666666666}{k}} \]
16. Add Preprocessing

Alternative 13: 33.9% accurate, 38.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\ell \cdot -0.6666666666666666}{t\_m \cdot k} \cdot \frac{\ell}{k}\right) \end{array} \]

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 -0.6666666666666666) (* t_m k)) (/ 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) {
	return t_s * (((l * -0.6666666666666666) / (t_m * k)) * (l / 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 * (-0.6666666666666666d0)) / (t_m * k)) * (l / 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 * -0.6666666666666666) / (t_m * k)) * (l / 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 * -0.6666666666666666) / (t_m * k)) * (l / k))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(Float64(l * -0.6666666666666666) / Float64(t_m * k)) * Float64(l / 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 * -0.6666666666666666) / (t_m * k)) * (l / k));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(N[(l * -0.6666666666666666), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}\right), \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k}\right), \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right), \tan k\right), \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right) \]

   5. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\sin k}\right), \tan k\right), \left(\left(\color{blue}{1} + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}\right), \sin k\right), \tan k\right), \left(\left(\color{blue}{1} + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{{t}^{3}}{\ell \cdot \ell}\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   8. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{\frac{{t}^{3}}{\ell}}{\ell}\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{t}^{3}}{\ell}\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({t}^{3}\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   11. cube-mult N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   14. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \ell\right)\right), \mathsf{sin.f64}\left(k\right)\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   15. tan-lowering-tan.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \ell\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \left(\left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right) + 1\right)\right) \]

3. Simplified 51.0%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{2}{\frac{\frac{t \cdot \left(t \cdot t\right)}{\ell}}{\ell}}}{\sin k}}{\tan k}}{2 + k \cdot \frac{k}{t \cdot t}}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{k \cdot {t}^{3}}\right)}, \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot {\ell}^{2}}{k \cdot {t}^{3}}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot {\ell}^{2}\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({t}^{3} \cdot k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   7. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\left(t \cdot {t}^{2}\right) \cdot k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(t \cdot \left({t}^{2} \cdot k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \left({t}^{2} \cdot k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left({t}^{2}\right), k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot t\right), k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 47.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

7. Simplified 47.2%

   \[\leadsto \frac{\frac{\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(t \cdot t\right) \cdot k\right)}}}{\tan k}}{2 + k \cdot \frac{k}{t \cdot t}} \]

8. Taylor expanded in k around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-2}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right)}, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + \frac{-2}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} \cdot \frac{-2}{3}}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   3. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + \left({k}^{2} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) \cdot \frac{-2}{3}}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + {k}^{2} \cdot \left(\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{-2}{3}\right)}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + {k}^{2} \cdot \left(\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right)}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + {k}^{2} \cdot \left(\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right)\right), \left({k}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

10. Simplified 22.7%

    \[\leadsto \frac{\color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)} + \left(k \cdot k\right) \cdot \frac{-0.6666666666666666 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)}}{k \cdot k}}}{2 + k \cdot \frac{k}{t \cdot t}} \]

11. Taylor expanded in k around inf

    \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
12. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{\frac{-2}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

    2. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2}{3} \cdot {\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \left({\ell}^{2}\right)\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \left(\ell \cdot \ell\right)\right), \left({k}^{\color{blue}{2}} \cdot t\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({k}^{\color{blue}{2}} \cdot t\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(t \cdot \color{blue}{{k}^{2}}\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \color{blue}{\left({k}^{2}\right)}\right)\right) \]

    8. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \left(k \cdot \color{blue}{k}\right)\right)\right) \]

    9. *-lowering-*.f64 26.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

13. Simplified 26.9%

    \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot k\right)}} \]
14. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \frac{\left(\frac{-2}{3} \cdot \ell\right) \cdot \ell}{\color{blue}{t} \cdot \left(k \cdot k\right)} \]

    2. associate-*r* N/A

       \[\leadsto \frac{\left(\frac{-2}{3} \cdot \ell\right) \cdot \ell}{\left(t \cdot k\right) \cdot \color{blue}{k}} \]

    3. times-frac N/A

       \[\leadsto \frac{\frac{-2}{3} \cdot \ell}{t \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-2}{3} \cdot \ell}{t \cdot k}\right), \color{blue}{\left(\frac{\ell}{k}\right)}\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{3} \cdot \ell\right), \left(t \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{k}\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \frac{-2}{3}\right), \left(t \cdot k\right)\right), \left(\frac{\ell}{k}\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \frac{-2}{3}\right), \left(t \cdot k\right)\right), \left(\frac{\ell}{k}\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \frac{-2}{3}\right), \mathsf{*.f64}\left(t, k\right)\right), \left(\frac{\ell}{k}\right)\right) \]

    9. /-lowering-/.f64 29.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \frac{-2}{3}\right), \mathsf{*.f64}\left(t, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{k}\right)\right) \]

15. Applied egg-rr 29.5%

    \[\leadsto \color{blue}{\frac{\ell \cdot -0.6666666666666666}{t \cdot k} \cdot \frac{\ell}{k}} \]
16. Add Preprocessing

Alternative 14: 33.0% accurate, 38.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot -0.6666666666666666\right) \cdot \frac{\ell}{t\_m \cdot \left(k \cdot k\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (* l -0.6666666666666666) (/ l (* 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) {
	return t_s * ((l * -0.6666666666666666) * (l / (t_m * (k * k))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l * (-0.6666666666666666d0)) * (l / (t_m * (k * k))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * -0.6666666666666666) * (l / (t_m * (k * k))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l * -0.6666666666666666) * (l / (t_m * (k * k))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l * -0.6666666666666666) * Float64(l / Float64(t_m * Float64(k * k)))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l * -0.6666666666666666) * (l / (t_m * (k * k))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l * -0.6666666666666666), $MachinePrecision] * N[(l / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}\right), \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k}\right), \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right), \tan k\right), \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right) \]

   5. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\sin k}\right), \tan k\right), \left(\left(\color{blue}{1} + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}\right), \sin k\right), \tan k\right), \left(\left(\color{blue}{1} + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{{t}^{3}}{\ell \cdot \ell}\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   8. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{\frac{{t}^{3}}{\ell}}{\ell}\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{t}^{3}}{\ell}\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({t}^{3}\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   11. cube-mult N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   14. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \ell\right)\right), \mathsf{sin.f64}\left(k\right)\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   15. tan-lowering-tan.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \ell\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \left(\left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right) + 1\right)\right) \]

3. Simplified 51.0%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{2}{\frac{\frac{t \cdot \left(t \cdot t\right)}{\ell}}{\ell}}}{\sin k}}{\tan k}}{2 + k \cdot \frac{k}{t \cdot t}}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{k \cdot {t}^{3}}\right)}, \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot {\ell}^{2}}{k \cdot {t}^{3}}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot {\ell}^{2}\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({t}^{3} \cdot k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   7. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\left(t \cdot {t}^{2}\right) \cdot k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(t \cdot \left({t}^{2} \cdot k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \left({t}^{2} \cdot k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left({t}^{2}\right), k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot t\right), k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 47.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

7. Simplified 47.2%

   \[\leadsto \frac{\frac{\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(t \cdot t\right) \cdot k\right)}}}{\tan k}}{2 + k \cdot \frac{k}{t \cdot t}} \]

8. Taylor expanded in k around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-2}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right)}, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + \frac{-2}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} \cdot \frac{-2}{3}}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   3. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + \left({k}^{2} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) \cdot \frac{-2}{3}}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + {k}^{2} \cdot \left(\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{-2}{3}\right)}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + {k}^{2} \cdot \left(\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right)}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + {k}^{2} \cdot \left(\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right)\right), \left({k}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

10. Simplified 22.7%

    \[\leadsto \frac{\color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)} + \left(k \cdot k\right) \cdot \frac{-0.6666666666666666 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)}}{k \cdot k}}}{2 + k \cdot \frac{k}{t \cdot t}} \]

11. Taylor expanded in k around inf

    \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
12. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{\frac{-2}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

    2. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2}{3} \cdot {\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \left({\ell}^{2}\right)\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \left(\ell \cdot \ell\right)\right), \left({k}^{\color{blue}{2}} \cdot t\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({k}^{\color{blue}{2}} \cdot t\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(t \cdot \color{blue}{{k}^{2}}\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \color{blue}{\left({k}^{2}\right)}\right)\right) \]

    8. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \left(k \cdot \color{blue}{k}\right)\right)\right) \]

    9. *-lowering-*.f64 26.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

13. Simplified 26.9%

    \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot k\right)}} \]
14. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \frac{\left(\frac{-2}{3} \cdot \ell\right) \cdot \ell}{\color{blue}{t} \cdot \left(k \cdot k\right)} \]

    2. associate-/l* N/A

       \[\leadsto \left(\frac{-2}{3} \cdot \ell\right) \cdot \color{blue}{\frac{\ell}{t \cdot \left(k \cdot k\right)}} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{-2}{3} \cdot \ell\right), \color{blue}{\left(\frac{\ell}{t \cdot \left(k \cdot k\right)}\right)}\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\ell \cdot \frac{-2}{3}\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(k \cdot k\right)}\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \frac{-2}{3}\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(k \cdot k\right)}\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \frac{-2}{3}\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \frac{-2}{3}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(k \cdot k\right)}\right)\right)\right) \]

    8. *-lowering-*.f64 28.1%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \frac{-2}{3}\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right) \]

15. Applied egg-rr 28.1%

    \[\leadsto \color{blue}{\left(\ell \cdot -0.6666666666666666\right) \cdot \frac{\ell}{t \cdot \left(k \cdot k\right)}} \]
16. Add Preprocessing

Alternative 15: 32.4% accurate, 38.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(-0.6666666666666666 \cdot \left(\ell \cdot \frac{\frac{\ell}{k \cdot k}}{t\_m}\right)\right) \end{array} \]

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 (* -0.6666666666666666 (* l (/ (/ l (* k k)) 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 * (-0.6666666666666666 * (l * ((l / (k * k)) / 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 * ((-0.6666666666666666d0) * (l * ((l / (k * k)) / 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 * (-0.6666666666666666 * (l * ((l / (k * k)) / 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 * (-0.6666666666666666 * (l * ((l / (k * k)) / 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(-0.6666666666666666 * Float64(l * Float64(Float64(l / Float64(k * k)) / 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 * (-0.6666666666666666 * (l * ((l / (k * k)) / 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[(-0.6666666666666666 * N[(l * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}\right), \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k}\right), \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right), \tan k\right), \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right) \]

   5. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\sin k}\right), \tan k\right), \left(\left(\color{blue}{1} + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}\right), \sin k\right), \tan k\right), \left(\left(\color{blue}{1} + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{{t}^{3}}{\ell \cdot \ell}\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   8. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{\frac{{t}^{3}}{\ell}}{\ell}\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{{t}^{3}}{\ell}\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({t}^{3}\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   11. cube-mult N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \ell\right)\right), \sin k\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   14. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \ell\right)\right), \mathsf{sin.f64}\left(k\right)\right), \tan k\right), \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \]

   15. tan-lowering-tan.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \ell\right)\right), \mathsf{sin.f64}\left(k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \left(\left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right) + 1\right)\right) \]

3. Simplified 51.0%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{2}{\frac{\frac{t \cdot \left(t \cdot t\right)}{\ell}}{\ell}}}{\sin k}}{\tan k}}{2 + k \cdot \frac{k}{t \cdot t}}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{k \cdot {t}^{3}}\right)}, \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot {\ell}^{2}}{k \cdot {t}^{3}}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot {\ell}^{2}\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(k \cdot {t}^{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({t}^{3} \cdot k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   7. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\left(t \cdot {t}^{2}\right) \cdot k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(t \cdot \left({t}^{2} \cdot k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \left({t}^{2} \cdot k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left({t}^{2}\right), k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot t\right), k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 47.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

7. Simplified 47.2%

   \[\leadsto \frac{\frac{\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(t \cdot t\right) \cdot k\right)}}}{\tan k}}{2 + k \cdot \frac{k}{t \cdot t}} \]

8. Taylor expanded in k around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-2}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right)}, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + \frac{-2}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} \cdot \frac{-2}{3}}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   3. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + \left({k}^{2} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) \cdot \frac{-2}{3}}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + {k}^{2} \cdot \left(\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{-2}{3}\right)}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + {k}^{2} \cdot \left(\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right)}{{k}^{2}}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{{t}^{3}} + {k}^{2} \cdot \left(\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right)\right), \left({k}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

10. Simplified 22.7%

    \[\leadsto \frac{\color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)} + \left(k \cdot k\right) \cdot \frac{-0.6666666666666666 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)}}{k \cdot k}}}{2 + k \cdot \frac{k}{t \cdot t}} \]

11. Taylor expanded in k around inf

    \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
12. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{\frac{-2}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

    2. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2}{3} \cdot {\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot t\right)}\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \left({\ell}^{2}\right)\right), \left(\color{blue}{{k}^{2}} \cdot t\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \left(\ell \cdot \ell\right)\right), \left({k}^{\color{blue}{2}} \cdot t\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({k}^{\color{blue}{2}} \cdot t\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(t \cdot \color{blue}{{k}^{2}}\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \color{blue}{\left({k}^{2}\right)}\right)\right) \]

    8. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \left(k \cdot \color{blue}{k}\right)\right)\right) \]

    9. *-lowering-*.f64 26.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

13. Simplified 26.9%

    \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot k\right)}} \]

14. Taylor expanded in l around 0

    \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
15. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-2}{3}, \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}\right) \]

    2. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-2}{3}, \left(\frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot t}\right)\right) \]

    3. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-2}{3}, \left(\ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}}\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{{k}^{2} \cdot t}\right)}\right)\right) \]

    5. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \left(\frac{\frac{\ell}{{k}^{2}}}{\color{blue}{t}}\right)\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\ell}{{k}^{2}}\right), \color{blue}{t}\right)\right)\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left({k}^{2}\right)\right), t\right)\right)\right) \]

    8. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), t\right)\right)\right) \]

    9. *-lowering-*.f64 27.7%

       \[\leadsto \mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), t\right)\right)\right) \]

16. Simplified 27.7%

    \[\leadsto \color{blue}{-0.6666666666666666 \cdot \left(\ell \cdot \frac{\frac{\ell}{k \cdot k}}{t}\right)} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024288 
(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))))