Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.7% → 88.8%

Time: 18.3s

Alternatives: 17

Speedup: 32.4×

• 26.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot k}{\ell} \cdot \left({\sin k}^{2} \cdot \frac{t\_m}{\ell}\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\frac{t\_m}{\frac{\ell}{t\_m \cdot \sin k}} \cdot \left(\tan k \cdot \left(2 + \frac{k}{\frac{t\_m \cdot t\_m}{k}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1e-76)
    (/ 2.0 (/ (* (/ (* k k) l) (* (pow (sin k) 2.0) (/ t_m l))) (cos k)))
    (/
     2.0
     (*
      (/ t_m l)
      (*
       (/ t_m (/ l (* t_m (sin k))))
       (* (tan k) (+ 2.0 (/ k (/ (* t_m t_m) k))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1e-76) {
		tmp = 2.0 / ((((k * k) / l) * (pow(sin(k), 2.0) * (t_m / l))) / cos(k));
	} else {
		tmp = 2.0 / ((t_m / l) * ((t_m / (l / (t_m * sin(k)))) * (tan(k) * (2.0 + (k / ((t_m * t_m) / k))))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1d-76) then
        tmp = 2.0d0 / ((((k * k) / l) * ((sin(k) ** 2.0d0) * (t_m / l))) / cos(k))
    else
        tmp = 2.0d0 / ((t_m / l) * ((t_m / (l / (t_m * sin(k)))) * (tan(k) * (2.0d0 + (k / ((t_m * t_m) / k))))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1e-76) {
		tmp = 2.0 / ((((k * k) / l) * (Math.pow(Math.sin(k), 2.0) * (t_m / l))) / Math.cos(k));
	} else {
		tmp = 2.0 / ((t_m / l) * ((t_m / (l / (t_m * Math.sin(k)))) * (Math.tan(k) * (2.0 + (k / ((t_m * t_m) / k))))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1e-76:
		tmp = 2.0 / ((((k * k) / l) * (math.pow(math.sin(k), 2.0) * (t_m / l))) / math.cos(k))
	else:
		tmp = 2.0 / ((t_m / l) * ((t_m / (l / (t_m * math.sin(k)))) * (math.tan(k) * (2.0 + (k / ((t_m * t_m) / k))))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1e-76)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) / l) * Float64((sin(k) ^ 2.0) * Float64(t_m / l))) / cos(k)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(t_m / Float64(l / Float64(t_m * sin(k)))) * Float64(tan(k) * Float64(2.0 + Float64(k / Float64(Float64(t_m * t_m) / k)))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1e-76)
		tmp = 2.0 / ((((k * k) / l) * ((sin(k) ^ 2.0) * (t_m / l))) / cos(k));
	else
		tmp = 2.0 / ((t_m / l) * ((t_m / (l / (t_m * sin(k)))) * (tan(k) * (2.0 + (k / ((t_m * t_m) / k))))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1e-76], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m / N[(l / N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[(k / N[(N[(t$95$m * t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.99999999999999927e-77

   1. Initial program 47.6%

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

      1. 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. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      11. sin-lowering-sin.f64 62.9%

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

   4. Applied egg-rr 62.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 67.3%

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

   6. Applied egg-rr 67.3%

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

   7. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

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

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      15. cos-lowering-cos.f64 68.5%

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

   9. Simplified 68.5%

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

   if 9.99999999999999927e-77 < t

   1. Initial program 69.0%

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

      1. 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. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      11. sin-lowering-sin.f64 82.6%

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

   4. Applied egg-rr 82.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 89.9%

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

   6. Applied egg-rr 89.9%

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

      1. associate-*l* N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. metadata-eval N/A

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

      9. unpow2 N/A

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

      10. clear-num N/A

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

      11. div-inv N/A

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

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

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

   8. Applied egg-rr 93.2%

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

Alternative 2: 78.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\frac{\ell}{t\_m \cdot \sin k}}{t\_m} \cdot \frac{1}{\frac{\tan k}{\frac{\ell}{t\_m}}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 1.06 \cdot 10^{+153}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t\_m \cdot t\_m}}{\tan k \cdot \frac{t\_m \cdot \left(2 + \frac{k}{\frac{t\_m \cdot t\_m}{k}}\right)}{\frac{\ell}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (/ (/ l (* t_m (sin k))) t_m) (/ 1.0 (/ (tan k) (/ l t_m))))))
   (*
    t_s
    (if (<= t_m 1.55e-162)
      t_2
      (if (<= t_m 1.06e+153)
        (*
         2.0
         (/
          (/ l (* t_m t_m))
          (*
           (tan k)
           (/ (* t_m (+ 2.0 (/ k (/ (* t_m t_m) k)))) (/ l (sin k))))))
        t_2)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = ((l / (t_m * sin(k))) / t_m) * (1.0 / (tan(k) / (l / t_m)));
	double tmp;
	if (t_m <= 1.55e-162) {
		tmp = t_2;
	} else if (t_m <= 1.06e+153) {
		tmp = 2.0 * ((l / (t_m * t_m)) / (tan(k) * ((t_m * (2.0 + (k / ((t_m * t_m) / k)))) / (l / sin(k)))));
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = ((l / (t_m * sin(k))) / t_m) * (1.0d0 / (tan(k) / (l / t_m)))
    if (t_m <= 1.55d-162) then
        tmp = t_2
    else if (t_m <= 1.06d+153) then
        tmp = 2.0d0 * ((l / (t_m * t_m)) / (tan(k) * ((t_m * (2.0d0 + (k / ((t_m * t_m) / k)))) / (l / sin(k)))))
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = ((l / (t_m * Math.sin(k))) / t_m) * (1.0 / (Math.tan(k) / (l / t_m)));
	double tmp;
	if (t_m <= 1.55e-162) {
		tmp = t_2;
	} else if (t_m <= 1.06e+153) {
		tmp = 2.0 * ((l / (t_m * t_m)) / (Math.tan(k) * ((t_m * (2.0 + (k / ((t_m * t_m) / k)))) / (l / Math.sin(k)))));
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = ((l / (t_m * math.sin(k))) / t_m) * (1.0 / (math.tan(k) / (l / t_m)))
	tmp = 0
	if t_m <= 1.55e-162:
		tmp = t_2
	elif t_m <= 1.06e+153:
		tmp = 2.0 * ((l / (t_m * t_m)) / (math.tan(k) * ((t_m * (2.0 + (k / ((t_m * t_m) / k)))) / (l / math.sin(k)))))
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(Float64(l / Float64(t_m * sin(k))) / t_m) * Float64(1.0 / Float64(tan(k) / Float64(l / t_m))))
	tmp = 0.0
	if (t_m <= 1.55e-162)
		tmp = t_2;
	elseif (t_m <= 1.06e+153)
		tmp = Float64(2.0 * Float64(Float64(l / Float64(t_m * t_m)) / Float64(tan(k) * Float64(Float64(t_m * Float64(2.0 + Float64(k / Float64(Float64(t_m * t_m) / k)))) / Float64(l / sin(k))))));
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = ((l / (t_m * sin(k))) / t_m) * (1.0 / (tan(k) / (l / t_m)));
	tmp = 0.0;
	if (t_m <= 1.55e-162)
		tmp = t_2;
	elseif (t_m <= 1.06e+153)
		tmp = 2.0 * ((l / (t_m * t_m)) / (tan(k) * ((t_m * (2.0 + (k / ((t_m * t_m) / k)))) / (l / sin(k)))));
	else
		tmp = t_2;
	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[(N[(l / N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(1.0 / N[(N[Tan[k], $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.55e-162], t$95$2, If[LessEqual[t$95$m, 1.06e+153], N[(2.0 * N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[(N[(t$95$m * N[(2.0 + N[(k / N[(N[(t$95$m * t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.5499999999999999e-162 or 1.05999999999999995e153 < t

   1. Initial program 53.1%

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

      1. 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. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      11. sin-lowering-sin.f64 66.6%

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

   4. Applied egg-rr 66.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 73.7%

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

   6. Applied egg-rr 73.7%

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

   7. Taylor expanded in k around 0

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

   9. Simplified 67.4%

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. clear-num N/A

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

       5. div-inv N/A

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

       6. associate-*l* N/A

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

       7. associate-/r* N/A

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

       8. metadata-eval N/A

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

       9. inv-pow N/A

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

       10. *-commutative N/A

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

       11. metadata-eval N/A

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

       12. unpow-prod-down N/A

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

   11. Applied egg-rr 72.0%

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

   if 1.5499999999999999e-162 < t < 1.05999999999999995e153

   1. Initial program 57.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 76.4%

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

   4. Applied egg-rr 76.4%

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

   6. Applied egg-rr 76.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

   8. Applied egg-rr 78.1%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot \sin k}}{t} \cdot \frac{1}{\frac{\tan k}{\frac{\ell}{t}}}\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{+153}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t \cdot t}}{\tan k \cdot \frac{t \cdot \left(2 + \frac{k}{\frac{t \cdot t}{k}}\right)}{\frac{\ell}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot \sin k}}{t} \cdot \frac{1}{\frac{\tan k}{\frac{\ell}{t}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\frac{\ell}{t\_m \cdot \sin k}}{t\_m} \cdot \frac{1}{\frac{\tan k}{\frac{\ell}{t\_m}}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.1 \cdot 10^{-165}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 4.6 \cdot 10^{+79}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{2}{t\_m}}{\left(t\_m \cdot t\_m\right) \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t\_m}}{t\_m}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (/ (/ l (* t_m (sin k))) t_m) (/ 1.0 (/ (tan k) (/ l t_m))))))
   (*
    t_s
    (if (<= t_m 1.1e-165)
      t_2
      (if (<= t_m 4.6e+79)
        (*
         l
         (*
          l
          (/
           (/ 2.0 t_m)
           (*
            (* t_m t_m)
            (* (+ 2.0 (/ (/ (* k k) t_m) t_m)) (* (sin k) (tan k)))))))
        t_2)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = ((l / (t_m * sin(k))) / t_m) * (1.0 / (tan(k) / (l / t_m)));
	double tmp;
	if (t_m <= 1.1e-165) {
		tmp = t_2;
	} else if (t_m <= 4.6e+79) {
		tmp = l * (l * ((2.0 / t_m) / ((t_m * t_m) * ((2.0 + (((k * k) / t_m) / t_m)) * (sin(k) * tan(k))))));
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = ((l / (t_m * sin(k))) / t_m) * (1.0d0 / (tan(k) / (l / t_m)))
    if (t_m <= 1.1d-165) then
        tmp = t_2
    else if (t_m <= 4.6d+79) then
        tmp = l * (l * ((2.0d0 / t_m) / ((t_m * t_m) * ((2.0d0 + (((k * k) / t_m) / t_m)) * (sin(k) * tan(k))))))
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = ((l / (t_m * Math.sin(k))) / t_m) * (1.0 / (Math.tan(k) / (l / t_m)));
	double tmp;
	if (t_m <= 1.1e-165) {
		tmp = t_2;
	} else if (t_m <= 4.6e+79) {
		tmp = l * (l * ((2.0 / t_m) / ((t_m * t_m) * ((2.0 + (((k * k) / t_m) / t_m)) * (Math.sin(k) * Math.tan(k))))));
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = ((l / (t_m * math.sin(k))) / t_m) * (1.0 / (math.tan(k) / (l / t_m)))
	tmp = 0
	if t_m <= 1.1e-165:
		tmp = t_2
	elif t_m <= 4.6e+79:
		tmp = l * (l * ((2.0 / t_m) / ((t_m * t_m) * ((2.0 + (((k * k) / t_m) / t_m)) * (math.sin(k) * math.tan(k))))))
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(Float64(l / Float64(t_m * sin(k))) / t_m) * Float64(1.0 / Float64(tan(k) / Float64(l / t_m))))
	tmp = 0.0
	if (t_m <= 1.1e-165)
		tmp = t_2;
	elseif (t_m <= 4.6e+79)
		tmp = Float64(l * Float64(l * Float64(Float64(2.0 / t_m) / Float64(Float64(t_m * t_m) * Float64(Float64(2.0 + Float64(Float64(Float64(k * k) / t_m) / t_m)) * Float64(sin(k) * tan(k)))))));
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = ((l / (t_m * sin(k))) / t_m) * (1.0 / (tan(k) / (l / t_m)));
	tmp = 0.0;
	if (t_m <= 1.1e-165)
		tmp = t_2;
	elseif (t_m <= 4.6e+79)
		tmp = l * (l * ((2.0 / t_m) / ((t_m * t_m) * ((2.0 + (((k * k) / t_m) / t_m)) * (sin(k) * tan(k))))));
	else
		tmp = t_2;
	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[(N[(l / N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(1.0 / N[(N[Tan[k], $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.1e-165], t$95$2, If[LessEqual[t$95$m, 4.6e+79], N[(l * N[(l * N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(N[(2.0 + N[(N[(N[(k * k), $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.0999999999999999e-165 or 4.6000000000000001e79 < t

   1. Initial program 52.4%

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

      1. 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. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      11. sin-lowering-sin.f64 69.0%

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

   4. Applied egg-rr 69.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 75.5%

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

   6. Applied egg-rr 75.5%

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

   7. Taylor expanded in k around 0

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

   9. Simplified 68.9%

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. clear-num N/A

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

       5. div-inv N/A

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

       6. associate-*l* N/A

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

       7. associate-/r* N/A

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

       8. metadata-eval N/A

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

       9. inv-pow N/A

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

       10. *-commutative N/A

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

       11. metadata-eval N/A

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

       12. unpow-prod-down N/A

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

   11. Applied egg-rr 73.1%

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

   if 1.0999999999999999e-165 < t < 4.6000000000000001e79

   1. Initial program 62.7%

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

   4. Applied egg-rr 75.3%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot \sin k}}{t} \cdot \frac{1}{\frac{\tan k}{\frac{\ell}{t}}}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+79}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{2}{t}}{\left(t \cdot t\right) \cdot \left(\left(2 + \frac{\frac{k \cdot k}{t}}{t}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot \sin k}}{t} \cdot \frac{1}{\frac{\tan k}{\frac{\ell}{t}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{t\_m \cdot \sin k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{+122}:\\ \;\;\;\;\frac{\frac{2}{\frac{\tan k}{\frac{\ell}{t\_m}}}}{\frac{t\_m \cdot \left(2 + \frac{k \cdot \frac{k}{t\_m}}{t\_m}\right)}{t\_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{t\_m}{t\_2} \cdot \left(\frac{t\_m}{\ell} \cdot \tan k\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ l (* t_m (sin k)))))
   (*
    t_s
    (if (<= l 5.4e+122)
      (/
       (/ 2.0 (/ (tan k) (/ l t_m)))
       (/ (* t_m (+ 2.0 (/ (* k (/ k t_m)) t_m))) t_2))
      (/ 2.0 (* 2.0 (* (/ t_m t_2) (* (/ t_m l) (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 t_2 = l / (t_m * sin(k));
	double tmp;
	if (l <= 5.4e+122) {
		tmp = (2.0 / (tan(k) / (l / t_m))) / ((t_m * (2.0 + ((k * (k / t_m)) / t_m))) / t_2);
	} else {
		tmp = 2.0 / (2.0 * ((t_m / t_2) * ((t_m / l) * 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) :: t_2
    real(8) :: tmp
    t_2 = l / (t_m * sin(k))
    if (l <= 5.4d+122) then
        tmp = (2.0d0 / (tan(k) / (l / t_m))) / ((t_m * (2.0d0 + ((k * (k / t_m)) / t_m))) / t_2)
    else
        tmp = 2.0d0 / (2.0d0 * ((t_m / t_2) * ((t_m / l) * 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 t_2 = l / (t_m * Math.sin(k));
	double tmp;
	if (l <= 5.4e+122) {
		tmp = (2.0 / (Math.tan(k) / (l / t_m))) / ((t_m * (2.0 + ((k * (k / t_m)) / t_m))) / t_2);
	} else {
		tmp = 2.0 / (2.0 * ((t_m / t_2) * ((t_m / l) * 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):
	t_2 = l / (t_m * math.sin(k))
	tmp = 0
	if l <= 5.4e+122:
		tmp = (2.0 / (math.tan(k) / (l / t_m))) / ((t_m * (2.0 + ((k * (k / t_m)) / t_m))) / t_2)
	else:
		tmp = 2.0 / (2.0 * ((t_m / t_2) * ((t_m / l) * 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)
	t_2 = Float64(l / Float64(t_m * sin(k)))
	tmp = 0.0
	if (l <= 5.4e+122)
		tmp = Float64(Float64(2.0 / Float64(tan(k) / Float64(l / t_m))) / Float64(Float64(t_m * Float64(2.0 + Float64(Float64(k * Float64(k / t_m)) / t_m))) / t_2));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(t_m / t_2) * Float64(Float64(t_m / l) * 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)
	t_2 = l / (t_m * sin(k));
	tmp = 0.0;
	if (l <= 5.4e+122)
		tmp = (2.0 / (tan(k) / (l / t_m))) / ((t_m * (2.0 + ((k * (k / t_m)) / t_m))) / t_2);
	else
		tmp = 2.0 / (2.0 * ((t_m / t_2) * ((t_m / l) * 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_] := Block[{t$95$2 = N[(l / N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l, 5.4e+122], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * N[(2.0 + N[(N[(k * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[(t$95$m / t$95$2), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.3999999999999997e122

   1. Initial program 58.7%

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

      1. 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. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      11. sin-lowering-sin.f64 72.4%

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

   4. Applied egg-rr 72.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 77.3%

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

   6. Applied egg-rr 77.3%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

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

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

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

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

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

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

      13. sin-lowering-sin.f64 79.4%

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

   8. Applied egg-rr 79.4%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

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

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

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

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

      10. associate-*l/ N/A

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

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

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

   10. Applied egg-rr 83.1%

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

   if 5.3999999999999997e122 < l

   1. Initial program 30.8%

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

      1. 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. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      11. sin-lowering-sin.f64 51.1%

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

   4. Applied egg-rr 51.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 58.4%

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

   6. Applied egg-rr 58.4%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

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

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

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

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

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

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

      13. sin-lowering-sin.f64 67.7%

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

   8. Applied egg-rr 67.7%

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

   9. Taylor expanded in k around 0

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

   11. Simplified 73.9%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{+122}:\\ \;\;\;\;\frac{\frac{2}{\frac{\tan k}{\frac{\ell}{t}}}}{\frac{t \cdot \left(2 + \frac{k \cdot \frac{k}{t}}{t}\right)}{\frac{\ell}{t \cdot \sin k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{t}{\frac{\ell}{t \cdot \sin k}} \cdot \left(\frac{t}{\ell} \cdot \tan k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{t\_m \cdot \sin k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.9 \cdot 10^{-162}:\\ \;\;\;\;\frac{t\_2}{t\_m} \cdot \frac{1}{\frac{\tan k}{\frac{\ell}{t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\frac{t\_m}{t\_2} \cdot \left(\tan k \cdot \left(2 + \frac{k}{\frac{t\_m \cdot t\_m}{k}}\right)\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ l (* t_m (sin k)))))
   (*
    t_s
    (if (<= t_m 1.9e-162)
      (* (/ t_2 t_m) (/ 1.0 (/ (tan k) (/ l t_m))))
      (/
       2.0
       (*
        (/ t_m l)
        (* (/ t_m t_2) (* (tan k) (+ 2.0 (/ k (/ (* t_m t_m) k)))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = l / (t_m * sin(k));
	double tmp;
	if (t_m <= 1.9e-162) {
		tmp = (t_2 / t_m) * (1.0 / (tan(k) / (l / t_m)));
	} else {
		tmp = 2.0 / ((t_m / l) * ((t_m / t_2) * (tan(k) * (2.0 + (k / ((t_m * t_m) / k))))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = l / (t_m * sin(k))
    if (t_m <= 1.9d-162) then
        tmp = (t_2 / t_m) * (1.0d0 / (tan(k) / (l / t_m)))
    else
        tmp = 2.0d0 / ((t_m / l) * ((t_m / t_2) * (tan(k) * (2.0d0 + (k / ((t_m * t_m) / k))))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = l / (t_m * Math.sin(k));
	double tmp;
	if (t_m <= 1.9e-162) {
		tmp = (t_2 / t_m) * (1.0 / (Math.tan(k) / (l / t_m)));
	} else {
		tmp = 2.0 / ((t_m / l) * ((t_m / t_2) * (Math.tan(k) * (2.0 + (k / ((t_m * t_m) / k))))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = l / (t_m * math.sin(k))
	tmp = 0
	if t_m <= 1.9e-162:
		tmp = (t_2 / t_m) * (1.0 / (math.tan(k) / (l / t_m)))
	else:
		tmp = 2.0 / ((t_m / l) * ((t_m / t_2) * (math.tan(k) * (2.0 + (k / ((t_m * t_m) / k))))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(l / Float64(t_m * sin(k)))
	tmp = 0.0
	if (t_m <= 1.9e-162)
		tmp = Float64(Float64(t_2 / t_m) * Float64(1.0 / Float64(tan(k) / Float64(l / t_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(t_m / t_2) * Float64(tan(k) * Float64(2.0 + Float64(k / Float64(Float64(t_m * 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 = l / (t_m * sin(k));
	tmp = 0.0;
	if (t_m <= 1.9e-162)
		tmp = (t_2 / t_m) * (1.0 / (tan(k) / (l / t_m)));
	else
		tmp = 2.0 / ((t_m / l) * ((t_m / t_2) * (tan(k) * (2.0 + (k / ((t_m * t_m) / k))))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(l / N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.9e-162], N[(N[(t$95$2 / t$95$m), $MachinePrecision] * N[(1.0 / N[(N[Tan[k], $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m / t$95$2), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[(k / N[(N[(t$95$m * t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.90000000000000002e-162

   1. Initial program 49.5%

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

      1. 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. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      11. sin-lowering-sin.f64 64.7%

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

   4. Applied egg-rr 64.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 69.6%

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

   6. Applied egg-rr 69.6%

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

   7. Taylor expanded in k around 0

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

   9. Simplified 62.0%

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. clear-num N/A

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

       5. div-inv N/A

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

       6. associate-*l* N/A

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

       7. associate-/r* N/A

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

       8. metadata-eval N/A

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

       9. inv-pow N/A

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

       10. *-commutative N/A

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

       11. metadata-eval N/A

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

       12. unpow-prod-down N/A

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

   11. Applied egg-rr 65.6%

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

   if 1.90000000000000002e-162 < t

   1. Initial program 62.0%

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

      1. 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. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      11. sin-lowering-sin.f64 76.0%

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

   4. Applied egg-rr 76.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 81.9%

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

   6. Applied egg-rr 81.9%

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

      1. associate-*l* N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. metadata-eval N/A

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

      9. unpow2 N/A

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

      10. clear-num N/A

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

      11. div-inv N/A

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

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

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

   8. Applied egg-rr 86.6%

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

Alternative 6: 75.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot \sin k}}{t\_m} \cdot \frac{1}{\frac{\tan k}{\frac{\ell}{t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \left(\tan k \cdot \frac{t\_m \cdot \left(2 + \frac{k}{\frac{t\_m \cdot t\_m}{k}}\right)}{\frac{\ell}{\sin k}}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.5e-31)
    (* (/ (/ l (* t_m (sin k))) t_m) (/ 1.0 (/ (tan k) (/ l t_m))))
    (/
     2.0
     (*
      (/ t_m l)
      (*
       t_m
       (*
        (tan k)
        (/ (* t_m (+ 2.0 (/ k (/ (* t_m t_m) k)))) (/ l (sin 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-31) {
		tmp = ((l / (t_m * sin(k))) / t_m) * (1.0 / (tan(k) / (l / t_m)));
	} else {
		tmp = 2.0 / ((t_m / l) * (t_m * (tan(k) * ((t_m * (2.0 + (k / ((t_m * t_m) / k)))) / (l / sin(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-31) then
        tmp = ((l / (t_m * sin(k))) / t_m) * (1.0d0 / (tan(k) / (l / t_m)))
    else
        tmp = 2.0d0 / ((t_m / l) * (t_m * (tan(k) * ((t_m * (2.0d0 + (k / ((t_m * t_m) / k)))) / (l / sin(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-31) {
		tmp = ((l / (t_m * Math.sin(k))) / t_m) * (1.0 / (Math.tan(k) / (l / t_m)));
	} else {
		tmp = 2.0 / ((t_m / l) * (t_m * (Math.tan(k) * ((t_m * (2.0 + (k / ((t_m * t_m) / k)))) / (l / Math.sin(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-31:
		tmp = ((l / (t_m * math.sin(k))) / t_m) * (1.0 / (math.tan(k) / (l / t_m)))
	else:
		tmp = 2.0 / ((t_m / l) * (t_m * (math.tan(k) * ((t_m * (2.0 + (k / ((t_m * t_m) / k)))) / (l / math.sin(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-31)
		tmp = Float64(Float64(Float64(l / Float64(t_m * sin(k))) / t_m) * Float64(1.0 / Float64(tan(k) / Float64(l / t_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(t_m * Float64(tan(k) * Float64(Float64(t_m * Float64(2.0 + Float64(k / Float64(Float64(t_m * t_m) / k)))) / Float64(l / sin(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-31)
		tmp = ((l / (t_m * sin(k))) / t_m) * (1.0 / (tan(k) / (l / t_m)));
	else
		tmp = 2.0 / ((t_m / l) * (t_m * (tan(k) * ((t_m * (2.0 + (k / ((t_m * t_m) / k)))) / (l / sin(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-31], N[(N[(N[(l / N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(1.0 / N[(N[Tan[k], $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[(N[Tan[k], $MachinePrecision] * N[(N[(t$95$m * N[(2.0 + N[(k / N[(N[(t$95$m * t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.49999999999999985e-31

   1. Initial program 56.0%

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

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

      4. associate-*l* N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      11. sin-lowering-sin.f64 69.9%

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

   4. Applied egg-rr 69.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 75.6%

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

   6. Applied egg-rr 75.6%

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

   7. Taylor expanded in k around 0

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

   9. Simplified 69.6%

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. clear-num N/A

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

       5. div-inv N/A

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

       6. associate-*l* N/A

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

       7. associate-/r* N/A

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

       8. metadata-eval N/A

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

       9. inv-pow N/A

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

       10. *-commutative N/A

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

       11. metadata-eval N/A

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

       12. unpow-prod-down N/A

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

   11. Applied egg-rr 75.2%

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

   if 3.49999999999999985e-31 < k

   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. 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 32.4%

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

   4. Applied egg-rr 32.4%

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

   6. Applied egg-rr 70.6%

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

      1. associate-*l/ N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. associate-*l* N/A

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

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

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

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

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

      10. associate-*l/ N/A

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

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

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

   8. Applied egg-rr 75.6%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot \sin k}}{t} \cdot \frac{1}{\frac{\tan k}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(t \cdot \left(\tan k \cdot \frac{t \cdot \left(2 + \frac{k}{\frac{t \cdot t}{k}}\right)}{\frac{\ell}{\sin k}}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{-149}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t\_m}}{t\_m \cdot t\_m}\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{2}{k}}{k \cdot \left(\frac{1}{\ell \cdot \ell} \cdot \left(2 \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right) + \left(k \cdot k\right) \cdot \left(t\_m \cdot \left(1 + \left(t\_m \cdot t\_m\right) \cdot 0.3333333333333333\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot \sin k}}{t\_m} \cdot \frac{1}{\frac{\tan k}{\frac{\ell}{t\_m}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 4e-149)
    (/ (* (/ l k) (/ (/ l k) t_m)) (* t_m t_m))
    (if (<= l 2.9e-74)
      (/
       (/ 2.0 k)
       (*
        k
        (*
         (/ 1.0 (* l l))
         (+
          (* 2.0 (* t_m (* t_m t_m)))
          (* (* k k) (* t_m (+ 1.0 (* (* t_m t_m) 0.3333333333333333))))))))
      (* (/ (/ l (* t_m (sin k))) t_m) (/ 1.0 (/ (tan k) (/ l t_m))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 4e-149) {
		tmp = ((l / k) * ((l / k) / t_m)) / (t_m * t_m);
	} else if (l <= 2.9e-74) {
		tmp = (2.0 / k) / (k * ((1.0 / (l * l)) * ((2.0 * (t_m * (t_m * t_m))) + ((k * k) * (t_m * (1.0 + ((t_m * t_m) * 0.3333333333333333)))))));
	} else {
		tmp = ((l / (t_m * sin(k))) / t_m) * (1.0 / (tan(k) / (l / t_m)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 4d-149) then
        tmp = ((l / k) * ((l / k) / t_m)) / (t_m * t_m)
    else if (l <= 2.9d-74) then
        tmp = (2.0d0 / k) / (k * ((1.0d0 / (l * l)) * ((2.0d0 * (t_m * (t_m * t_m))) + ((k * k) * (t_m * (1.0d0 + ((t_m * t_m) * 0.3333333333333333d0)))))))
    else
        tmp = ((l / (t_m * sin(k))) / t_m) * (1.0d0 / (tan(k) / (l / t_m)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 4e-149) {
		tmp = ((l / k) * ((l / k) / t_m)) / (t_m * t_m);
	} else if (l <= 2.9e-74) {
		tmp = (2.0 / k) / (k * ((1.0 / (l * l)) * ((2.0 * (t_m * (t_m * t_m))) + ((k * k) * (t_m * (1.0 + ((t_m * t_m) * 0.3333333333333333)))))));
	} else {
		tmp = ((l / (t_m * Math.sin(k))) / t_m) * (1.0 / (Math.tan(k) / (l / t_m)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 4e-149:
		tmp = ((l / k) * ((l / k) / t_m)) / (t_m * t_m)
	elif l <= 2.9e-74:
		tmp = (2.0 / k) / (k * ((1.0 / (l * l)) * ((2.0 * (t_m * (t_m * t_m))) + ((k * k) * (t_m * (1.0 + ((t_m * t_m) * 0.3333333333333333)))))))
	else:
		tmp = ((l / (t_m * math.sin(k))) / t_m) * (1.0 / (math.tan(k) / (l / t_m)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 4e-149)
		tmp = Float64(Float64(Float64(l / k) * Float64(Float64(l / k) / t_m)) / Float64(t_m * t_m));
	elseif (l <= 2.9e-74)
		tmp = Float64(Float64(2.0 / k) / Float64(k * Float64(Float64(1.0 / Float64(l * l)) * Float64(Float64(2.0 * Float64(t_m * Float64(t_m * t_m))) + Float64(Float64(k * k) * Float64(t_m * Float64(1.0 + Float64(Float64(t_m * t_m) * 0.3333333333333333))))))));
	else
		tmp = Float64(Float64(Float64(l / Float64(t_m * sin(k))) / t_m) * Float64(1.0 / Float64(tan(k) / Float64(l / t_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 4e-149)
		tmp = ((l / k) * ((l / k) / t_m)) / (t_m * t_m);
	elseif (l <= 2.9e-74)
		tmp = (2.0 / k) / (k * ((1.0 / (l * l)) * ((2.0 * (t_m * (t_m * t_m))) + ((k * k) * (t_m * (1.0 + ((t_m * t_m) * 0.3333333333333333)))))));
	else
		tmp = ((l / (t_m * sin(k))) / t_m) * (1.0 / (tan(k) / (l / t_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 4e-149], N[(N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.9e-74], N[(N[(2.0 / k), $MachinePrecision] / N[(k * N[(N[(1.0 / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(k * k), $MachinePrecision] * N[(t$95$m * N[(1.0 + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(1.0 / N[(N[Tan[k], $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if l < 3.99999999999999992e-149

   1. Initial program 57.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 55.0%

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

   5. Simplified 55.0%

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

      1. associate-/r* N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. *-lowering-*.f64 59.7%

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

   7. Applied egg-rr 59.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 65.8%

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

   9. Applied egg-rr 65.8%

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

   if 3.99999999999999992e-149 < l < 2.9e-74

   1. Initial program 59.4%

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

   3. Taylor expanded in k around 0

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

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

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

      2. unpow2 N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

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

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

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

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

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

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

   5. Simplified 83.1%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. div-inv N/A

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

      7. div-inv N/A

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

      8. distribute-rgt-out N/A

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

   7. Applied egg-rr 88.7%

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

   if 2.9e-74 < l

   1. Initial program 46.0%

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

      1. 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. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      11. sin-lowering-sin.f64 59.9%

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

   4. Applied egg-rr 59.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 64.8%

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

   6. Applied egg-rr 64.8%

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

   7. Taylor expanded in k around 0

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

   9. Simplified 65.6%

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. clear-num N/A

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

       5. div-inv N/A

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

       6. associate-*l* N/A

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

       7. associate-/r* N/A

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

       8. metadata-eval N/A

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

       9. inv-pow N/A

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

       10. *-commutative N/A

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

       11. metadata-eval N/A

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

       12. unpow-prod-down N/A

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

   11. Applied egg-rr 71.9%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{t \cdot \sin k}}{t} \cdot \frac{1}{\frac{\tan k}{\frac{\ell}{t}}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 67.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 4.8 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t\_m}}{t\_m \cdot t\_m}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{2}{k}}{k \cdot \left(\frac{1}{\ell \cdot \ell} \cdot \left(2 \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right) + \left(k \cdot k\right) \cdot \left(t\_m \cdot \left(1 + \left(t\_m \cdot t\_m\right) \cdot 0.3333333333333333\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(\left(t\_m \cdot \frac{t\_m}{\ell}\right) \cdot \left(k \cdot \left(\frac{t\_m}{\ell} + -0.16666666666666666 \cdot \frac{t\_m \cdot \left(k \cdot k\right)}{\ell}\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 4.8e-148)
    (/ (* (/ l k) (/ (/ l k) t_m)) (* t_m t_m))
    (if (<= l 5e-74)
      (/
       (/ 2.0 k)
       (*
        k
        (*
         (/ 1.0 (* l l))
         (+
          (* 2.0 (* t_m (* t_m t_m)))
          (* (* k k) (* t_m (+ 1.0 (* (* t_m t_m) 0.3333333333333333))))))))
      (/
       2.0
       (*
        2.0
        (*
         (tan k)
         (*
          (* t_m (/ t_m l))
          (*
           k
           (+
            (/ t_m l)
            (* -0.16666666666666666 (/ (* t_m (* k k)) l))))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 4.8e-148) {
		tmp = ((l / k) * ((l / k) / t_m)) / (t_m * t_m);
	} else if (l <= 5e-74) {
		tmp = (2.0 / k) / (k * ((1.0 / (l * l)) * ((2.0 * (t_m * (t_m * t_m))) + ((k * k) * (t_m * (1.0 + ((t_m * t_m) * 0.3333333333333333)))))));
	} else {
		tmp = 2.0 / (2.0 * (tan(k) * ((t_m * (t_m / l)) * (k * ((t_m / l) + (-0.16666666666666666 * ((t_m * (k * k)) / l)))))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 4.8d-148) then
        tmp = ((l / k) * ((l / k) / t_m)) / (t_m * t_m)
    else if (l <= 5d-74) then
        tmp = (2.0d0 / k) / (k * ((1.0d0 / (l * l)) * ((2.0d0 * (t_m * (t_m * t_m))) + ((k * k) * (t_m * (1.0d0 + ((t_m * t_m) * 0.3333333333333333d0)))))))
    else
        tmp = 2.0d0 / (2.0d0 * (tan(k) * ((t_m * (t_m / l)) * (k * ((t_m / l) + ((-0.16666666666666666d0) * ((t_m * (k * k)) / l)))))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 4.8e-148) {
		tmp = ((l / k) * ((l / k) / t_m)) / (t_m * t_m);
	} else if (l <= 5e-74) {
		tmp = (2.0 / k) / (k * ((1.0 / (l * l)) * ((2.0 * (t_m * (t_m * t_m))) + ((k * k) * (t_m * (1.0 + ((t_m * t_m) * 0.3333333333333333)))))));
	} else {
		tmp = 2.0 / (2.0 * (Math.tan(k) * ((t_m * (t_m / l)) * (k * ((t_m / l) + (-0.16666666666666666 * ((t_m * (k * k)) / l)))))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 4.8e-148:
		tmp = ((l / k) * ((l / k) / t_m)) / (t_m * t_m)
	elif l <= 5e-74:
		tmp = (2.0 / k) / (k * ((1.0 / (l * l)) * ((2.0 * (t_m * (t_m * t_m))) + ((k * k) * (t_m * (1.0 + ((t_m * t_m) * 0.3333333333333333)))))))
	else:
		tmp = 2.0 / (2.0 * (math.tan(k) * ((t_m * (t_m / l)) * (k * ((t_m / l) + (-0.16666666666666666 * ((t_m * (k * k)) / l)))))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 4.8e-148)
		tmp = Float64(Float64(Float64(l / k) * Float64(Float64(l / k) / t_m)) / Float64(t_m * t_m));
	elseif (l <= 5e-74)
		tmp = Float64(Float64(2.0 / k) / Float64(k * Float64(Float64(1.0 / Float64(l * l)) * Float64(Float64(2.0 * Float64(t_m * Float64(t_m * t_m))) + Float64(Float64(k * k) * Float64(t_m * Float64(1.0 + Float64(Float64(t_m * t_m) * 0.3333333333333333))))))));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(tan(k) * Float64(Float64(t_m * Float64(t_m / l)) * Float64(k * Float64(Float64(t_m / l) + Float64(-0.16666666666666666 * Float64(Float64(t_m * Float64(k * k)) / l))))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 4.8e-148)
		tmp = ((l / k) * ((l / k) / t_m)) / (t_m * t_m);
	elseif (l <= 5e-74)
		tmp = (2.0 / k) / (k * ((1.0 / (l * l)) * ((2.0 * (t_m * (t_m * t_m))) + ((k * k) * (t_m * (1.0 + ((t_m * t_m) * 0.3333333333333333)))))));
	else
		tmp = 2.0 / (2.0 * (tan(k) * ((t_m * (t_m / l)) * (k * ((t_m / l) + (-0.16666666666666666 * ((t_m * (k * k)) / l)))))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 4.8e-148], N[(N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5e-74], N[(N[(2.0 / k), $MachinePrecision] / N[(k * N[(N[(1.0 / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(k * k), $MachinePrecision] * N[(t$95$m * N[(1.0 + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[Tan[k], $MachinePrecision] * N[(N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(k * N[(N[(t$95$m / l), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if l < 4.8000000000000002e-148

   1. Initial program 57.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 55.0%

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

   5. Simplified 55.0%

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

      1. associate-/r* N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. *-lowering-*.f64 59.7%

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

   7. Applied egg-rr 59.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 65.8%

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

   9. Applied egg-rr 65.8%

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

   if 4.8000000000000002e-148 < l < 4.99999999999999998e-74

   1. Initial program 59.4%

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

   3. Taylor expanded in k around 0

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

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

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

      2. unpow2 N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

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

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

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

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

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

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

   5. Simplified 83.1%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. div-inv N/A

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

      7. div-inv N/A

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

      8. distribute-rgt-out N/A

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

   7. Applied egg-rr 88.7%

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

   if 4.99999999999999998e-74 < l

   1. Initial program 46.0%

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

      1. 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. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      11. sin-lowering-sin.f64 59.9%

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

   4. Applied egg-rr 59.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 64.8%

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

   6. Applied egg-rr 64.8%

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

   7. Taylor expanded in k around 0

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

   9. Simplified 65.6%

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

   10. Taylor expanded in k around 0

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), t\right), \color{blue}{\left(k \cdot \left(\frac{-1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell} + \frac{t}{\ell}\right)\right)}\right), \mathsf{tan.f64}\left(k\right)\right), 2\right)\right) \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), t\right), \left(k \cdot \left(\frac{t}{\ell} + \frac{-1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell}\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), 2\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), t\right), \left(k \cdot \left(\frac{t}{\ell} + \frac{{k}^{2} \cdot t}{\ell} \cdot \frac{-1}{6}\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), 2\right)\right) \]

       3. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), t\right), \left(k \cdot \left(\frac{t}{\ell} + \left({k}^{2} \cdot \frac{t}{\ell}\right) \cdot \frac{-1}{6}\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), 2\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), t\right), \left(k \cdot \left(\frac{t}{\ell} + {k}^{2} \cdot \left(\frac{t}{\ell} \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), 2\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), t\right), \left(k \cdot \left(\frac{t}{\ell} + {k}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{t}{\ell}\right)\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), t\right), \mathsf{*.f64}\left(k, \left(\frac{t}{\ell} + {k}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{t}{\ell}\right)\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), t\right), \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\left(\frac{t}{\ell}\right), \left({k}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{t}{\ell}\right)\right)\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), 2\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), t\right), \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left({k}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{t}{\ell}\right)\right)\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), 2\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), t\right), \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left({k}^{2} \cdot \left(\frac{t}{\ell} \cdot \frac{-1}{6}\right)\right)\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), 2\right)\right) \]

       10. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), t\right), \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\left({k}^{2} \cdot \frac{t}{\ell}\right) \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), 2\right)\right) \]

       11. associate-/l* N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), t\right), \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{{k}^{2} \cdot t}{\ell} \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), 2\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), t\right), \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{-1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell}\right)\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), 2\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), t\right), \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{*.f64}\left(\frac{-1}{6}, \left(\frac{{k}^{2} \cdot t}{\ell}\right)\right)\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), 2\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), t\right), \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{/.f64}\left(\left({k}^{2} \cdot t\right), \ell\right)\right)\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), 2\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), t\right), \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), t\right), \ell\right)\right)\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), 2\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), t\right), \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), t\right), \ell\right)\right)\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), 2\right)\right) \]

       17. *-lowering-*.f64 52.7%

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), t\right), \mathsf{*.f64}\left(k, \mathsf{+.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right)\right)\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), 2\right)\right) \]

   12. Simplified 52.7%

       \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(\frac{t}{\ell} + -0.16666666666666666 \cdot \frac{\left(k \cdot k\right) \cdot t}{\ell}\right)\right)}\right) \cdot \tan k\right) \cdot 2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.8 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}}{t \cdot t}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{2}{k}}{k \cdot \left(\frac{1}{\ell \cdot \ell} \cdot \left(2 \cdot \left(t \cdot \left(t \cdot t\right)\right) + \left(k \cdot k\right) \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot 0.3333333333333333\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot \left(\frac{t}{\ell} + -0.16666666666666666 \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t\_m}}{t\_m \cdot t\_m}\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{+61}:\\ \;\;\;\;\frac{\frac{2}{k}}{k \cdot \left(\frac{1}{\ell \cdot \ell} \cdot \left(2 \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right) + \left(k \cdot k\right) \cdot \left(t\_m \cdot \left(1 + \left(t\_m \cdot t\_m\right) \cdot 0.3333333333333333\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(\left(t\_m \cdot \frac{t\_m}{\ell}\right) \cdot \frac{t\_m \cdot k}{\ell}\right)\right)}\\ \end{array} \end{array} \]

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 3e-154)
    (/ (* (/ l k) (/ (/ l k) t_m)) (* t_m t_m))
    (if (<= l 2.05e+61)
      (/
       (/ 2.0 k)
       (*
        k
        (*
         (/ 1.0 (* l l))
         (+
          (* 2.0 (* t_m (* t_m t_m)))
          (* (* k k) (* t_m (+ 1.0 (* (* t_m t_m) 0.3333333333333333))))))))
      (/ 2.0 (* 2.0 (* (tan k) (* (* t_m (/ t_m l)) (/ (* t_m k) l)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 3e-154) {
		tmp = ((l / k) * ((l / k) / t_m)) / (t_m * t_m);
	} else if (l <= 2.05e+61) {
		tmp = (2.0 / k) / (k * ((1.0 / (l * l)) * ((2.0 * (t_m * (t_m * t_m))) + ((k * k) * (t_m * (1.0 + ((t_m * t_m) * 0.3333333333333333)))))));
	} else {
		tmp = 2.0 / (2.0 * (tan(k) * ((t_m * (t_m / l)) * ((t_m * k) / l))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 3d-154) then
        tmp = ((l / k) * ((l / k) / t_m)) / (t_m * t_m)
    else if (l <= 2.05d+61) then
        tmp = (2.0d0 / k) / (k * ((1.0d0 / (l * l)) * ((2.0d0 * (t_m * (t_m * t_m))) + ((k * k) * (t_m * (1.0d0 + ((t_m * t_m) * 0.3333333333333333d0)))))))
    else
        tmp = 2.0d0 / (2.0d0 * (tan(k) * ((t_m * (t_m / l)) * ((t_m * k) / l))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 3e-154) {
		tmp = ((l / k) * ((l / k) / t_m)) / (t_m * t_m);
	} else if (l <= 2.05e+61) {
		tmp = (2.0 / k) / (k * ((1.0 / (l * l)) * ((2.0 * (t_m * (t_m * t_m))) + ((k * k) * (t_m * (1.0 + ((t_m * t_m) * 0.3333333333333333)))))));
	} else {
		tmp = 2.0 / (2.0 * (Math.tan(k) * ((t_m * (t_m / l)) * ((t_m * k) / l))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 3e-154:
		tmp = ((l / k) * ((l / k) / t_m)) / (t_m * t_m)
	elif l <= 2.05e+61:
		tmp = (2.0 / k) / (k * ((1.0 / (l * l)) * ((2.0 * (t_m * (t_m * t_m))) + ((k * k) * (t_m * (1.0 + ((t_m * t_m) * 0.3333333333333333)))))))
	else:
		tmp = 2.0 / (2.0 * (math.tan(k) * ((t_m * (t_m / l)) * ((t_m * k) / l))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 3e-154)
		tmp = Float64(Float64(Float64(l / k) * Float64(Float64(l / k) / t_m)) / Float64(t_m * t_m));
	elseif (l <= 2.05e+61)
		tmp = Float64(Float64(2.0 / k) / Float64(k * Float64(Float64(1.0 / Float64(l * l)) * Float64(Float64(2.0 * Float64(t_m * Float64(t_m * t_m))) + Float64(Float64(k * k) * Float64(t_m * Float64(1.0 + Float64(Float64(t_m * t_m) * 0.3333333333333333))))))));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(tan(k) * Float64(Float64(t_m * Float64(t_m / l)) * Float64(Float64(t_m * k) / l)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 3e-154)
		tmp = ((l / k) * ((l / k) / t_m)) / (t_m * t_m);
	elseif (l <= 2.05e+61)
		tmp = (2.0 / k) / (k * ((1.0 / (l * l)) * ((2.0 * (t_m * (t_m * t_m))) + ((k * k) * (t_m * (1.0 + ((t_m * t_m) * 0.3333333333333333)))))));
	else
		tmp = 2.0 / (2.0 * (tan(k) * ((t_m * (t_m / l)) * ((t_m * k) / l))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 3e-154], N[(N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.05e+61], N[(N[(2.0 / k), $MachinePrecision] / N[(k * N[(N[(1.0 / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(k * k), $MachinePrecision] * N[(t$95$m * N[(1.0 + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[Tan[k], $MachinePrecision] * N[(N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if l < 3.0000000000000002e-154

   1. Initial program 57.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 55.0%

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

   5. Simplified 55.0%

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

      1. associate-/r* N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. *-lowering-*.f64 59.7%

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

   7. Applied egg-rr 59.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 65.8%

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

   9. Applied egg-rr 65.8%

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

   if 3.0000000000000002e-154 < l < 2.04999999999999986e61

   1. Initial program 58.6%

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

   3. Taylor expanded in k around 0

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

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

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

      2. unpow2 N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

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

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

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

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

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

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

   5. Simplified 71.1%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. div-inv N/A

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

      7. div-inv N/A

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

      8. distribute-rgt-out N/A

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

   7. Applied egg-rr 73.3%

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

   if 2.04999999999999986e61 < l

   1. Initial program 40.2%

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

      1. 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. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      11. sin-lowering-sin.f64 57.2%

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

   4. Applied egg-rr 57.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 64.5%

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

   6. Applied egg-rr 64.5%

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

   7. Taylor expanded in k around 0

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

   9. Simplified 67.3%

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

   10. Taylor expanded in k around 0

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

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

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

       2. *-lowering-*.f64 61.1%

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

   12. Simplified 61.1%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}}{t \cdot t}\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{+61}:\\ \;\;\;\;\frac{\frac{2}{k}}{k \cdot \left(\frac{1}{\ell \cdot \ell} \cdot \left(2 \cdot \left(t \cdot \left(t \cdot t\right)\right) + \left(k \cdot k\right) \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot 0.3333333333333333\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \frac{t \cdot k}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.2% accurate, 18.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.9 \cdot 10^{-48}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t\_m \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}\\ \mathbf{elif}\;t\_m \leq 6.8 \cdot 10^{+129}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot t\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.9e-48)
    (/ 2.0 (* (* k k) (/ (* t_m (* k k)) (* l l))))
    (if (<= t_m 6.8e+129)
      (* (/ l k) (/ l (* t_m (* k (* t_m t_m)))))
      (* l (/ l (* t_m (* k (* t_m (* t_m k))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.9e-48) {
		tmp = 2.0 / ((k * k) * ((t_m * (k * k)) / (l * l)));
	} else if (t_m <= 6.8e+129) {
		tmp = (l / k) * (l / (t_m * (k * (t_m * t_m))));
	} else {
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.9d-48) then
        tmp = 2.0d0 / ((k * k) * ((t_m * (k * k)) / (l * l)))
    else if (t_m <= 6.8d+129) then
        tmp = (l / k) * (l / (t_m * (k * (t_m * t_m))))
    else
        tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.9e-48) {
		tmp = 2.0 / ((k * k) * ((t_m * (k * k)) / (l * l)));
	} else if (t_m <= 6.8e+129) {
		tmp = (l / k) * (l / (t_m * (k * (t_m * t_m))));
	} else {
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.9e-48:
		tmp = 2.0 / ((k * k) * ((t_m * (k * k)) / (l * l)))
	elif t_m <= 6.8e+129:
		tmp = (l / k) * (l / (t_m * (k * (t_m * t_m))))
	else:
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.9e-48)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(t_m * Float64(k * k)) / Float64(l * l))));
	elseif (t_m <= 6.8e+129)
		tmp = Float64(Float64(l / k) * Float64(l / Float64(t_m * Float64(k * Float64(t_m * t_m)))));
	else
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(k * Float64(t_m * Float64(t_m * k))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.9e-48)
		tmp = 2.0 / ((k * k) * ((t_m * (k * k)) / (l * l)));
	elseif (t_m <= 6.8e+129)
		tmp = (l / k) * (l / (t_m * (k * (t_m * t_m))));
	else
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.9e-48], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.8e+129], N[(N[(l / k), $MachinePrecision] * N[(l / N[(t$95$m * N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(t$95$m * N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.90000000000000001e-48

   1. Initial program 48.2%

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

   3. Taylor expanded in k around 0

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

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

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

      2. unpow2 N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

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

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

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

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

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

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

   5. Simplified 49.5%

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

   6. Taylor expanded in t around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 52.4%

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

   8. Simplified 52.4%

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

   if 1.90000000000000001e-48 < t < 6.80000000000000036e129

   1. Initial program 69.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 64.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 64.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right) \cdot \color{blue}{k}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)} \cdot \color{blue}{\frac{\ell}{k}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right), \color{blue}{\left(\frac{\ell}{k}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\color{blue}{\ell}}{k}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(t \cdot t\right) \cdot k\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot t\right), k\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      10. /-lowering-/.f64 79.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{k}\right)\right) \]

   7. Applied egg-rr 79.8%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot k\right)} \cdot \frac{\ell}{k}} \]

   if 6.80000000000000036e129 < t

   1. Initial program 66.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 66.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 66.5%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{t}}\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{\ell}{t \cdot t}}{\color{blue}{t}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot t}\right), \color{blue}{t}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot t\right)\right), t\right)\right) \]

      10. *-lowering-*.f64 66.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right), t\right)\right) \]

   7. Applied egg-rr 66.8%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\frac{\ell}{t \cdot t}}{t}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{\ell}{t \cdot t}}{t} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

      2. associate-/l/ N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot t\right)} \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]

      3. frac-times N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot \color{blue}{k}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right) \cdot k} \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \]

      7. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \color{blue}{\left(k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)\right)}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right) \cdot \color{blue}{k}\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot k\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot k\right)}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(\left(t \cdot t\right) \cdot k\right), \color{blue}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot k\right)\right), k\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot \left(k \cdot t\right)\right), k\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot t\right)\right), k\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot k\right)\right), k\right)\right)\right)\right) \]

      18. *-lowering-*.f64 89.3%

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, k\right)\right), k\right)\right)\right)\right) \]

   9. Applied egg-rr 89.3%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot \left(t \cdot k\right)\right) \cdot k\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.9 \cdot 10^{-48}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+129}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.0% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.95 \cdot 10^{+58}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\frac{\ell \cdot \ell}{k}}{t\_m}}{k}}{t\_m}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.95e+58)
    (* l (/ l (* t_m (* k (* t_m (* t_m k))))))
    (/ (/ (/ (/ (/ (* l l) k) t_m) k) t_m) t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.95e+58) {
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))));
	} else {
		tmp = (((((l * l) / k) / t_m) / k) / t_m) / t_m;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.95d+58) then
        tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))))
    else
        tmp = (((((l * l) / k) / t_m) / k) / t_m) / t_m
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.95e+58) {
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))));
	} else {
		tmp = (((((l * l) / k) / t_m) / k) / t_m) / t_m;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 3.95e+58:
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))))
	else:
		tmp = (((((l * l) / k) / t_m) / k) / t_m) / t_m
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3.95e+58)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(k * Float64(t_m * Float64(t_m * k))))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(l * l) / k) / t_m) / k) / t_m) / t_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 3.95e+58)
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))));
	else
		tmp = (((((l * l) / k) / t_m) / k) / t_m) / t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 3.95e+58], N[(l * N[(l / N[(t$95$m * N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(l * l), $MachinePrecision] / k), $MachinePrecision] / t$95$m), $MachinePrecision] / k), $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.94999999999999994e58

   1. Initial program 56.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 52.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 52.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{t}}\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{\ell}{t \cdot t}}{\color{blue}{t}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot t}\right), \color{blue}{t}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot t\right)\right), t\right)\right) \]

      10. *-lowering-*.f64 55.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right), t\right)\right) \]

   7. Applied egg-rr 55.4%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\frac{\ell}{t \cdot t}}{t}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{\ell}{t \cdot t}}{t} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

      2. associate-/l/ N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot t\right)} \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]

      3. frac-times N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot \color{blue}{k}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right) \cdot k} \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \]

      7. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \color{blue}{\left(k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)\right)}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right) \cdot \color{blue}{k}\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot k\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot k\right)}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(\left(t \cdot t\right) \cdot k\right), \color{blue}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot k\right)\right), k\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot \left(k \cdot t\right)\right), k\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot t\right)\right), k\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot k\right)\right), k\right)\right)\right)\right) \]

      18. *-lowering-*.f64 64.5%

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, k\right)\right), k\right)\right)\right)\right) \]

   9. Applied egg-rr 64.5%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot \left(t \cdot k\right)\right) \cdot k\right)}} \]

   if 3.94999999999999994e58 < k

   1. Initial program 47.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 47.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 47.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{t}}\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{\ell}{t \cdot t}}{\color{blue}{t}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot t}\right), \color{blue}{t}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot t\right)\right), t\right)\right) \]

      10. *-lowering-*.f64 57.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right), t\right)\right) \]

   7. Applied egg-rr 57.5%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\frac{\ell}{t \cdot t}}{t}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{\ell}{t \cdot t}}{t} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

      2. associate-/l/ N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot t\right)} \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]

      3. frac-times N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot \color{blue}{k}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right) \cdot k} \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \]

      7. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \color{blue}{\left(k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)\right)}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right) \cdot \color{blue}{k}\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot k\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot k\right)}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(\left(t \cdot t\right) \cdot k\right), \color{blue}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot k\right)\right), k\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot \left(k \cdot t\right)\right), k\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot t\right)\right), k\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot k\right)\right), k\right)\right)\right)\right) \]

      18. *-lowering-*.f64 54.9%

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, k\right)\right), k\right)\right)\right)\right) \]

   9. Applied egg-rr 54.9%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot \left(t \cdot k\right)\right) \cdot k\right)}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot \left(t \cdot k\right)\right) \cdot k\right)}} \]

       2. associate-*r* N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot \color{blue}{k}} \]

       3. associate-/l/ N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{k}}{\color{blue}{t \cdot \left(t \cdot \left(t \cdot k\right)\right)}} \]

       4. associate-*r* N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{k}}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

       5. associate-/l/ N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{k}}{t \cdot k}}{\color{blue}{t \cdot t}} \]

       6. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\frac{\ell \cdot \ell}{k}}{t \cdot k}}{t}}{\color{blue}{t}} \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\ell \cdot \ell}{k}}{t \cdot k}}{t}\right), \color{blue}{t}\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\ell \cdot \ell}{k}}{t \cdot k}\right), t\right), t\right) \]

       9. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\frac{\ell \cdot \ell}{k}}{t}}{k}\right), t\right), t\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\ell \cdot \ell}{k}}{t}\right), k\right), t\right), t\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell \cdot \ell}{k}\right), t\right), k\right), t\right), t\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), k\right), t\right), k\right), t\right), t\right) \]

       13. *-lowering-*.f64 63.4%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), k\right), t\right), k\right), t\right), t\right) \]

   11. Applied egg-rr 63.4%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{\ell \cdot \ell}{k}}{t}}{k}}{t}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.95 \cdot 10^{+58}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\frac{\ell \cdot \ell}{k}}{t}}{k}}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.3% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{k}}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t\_m}}{t\_m}}{k \cdot \left(t\_m \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 2.5e-10)
    (/ (* l (/ l k)) (* t_m (* t_m (* t_m k))))
    (* l (/ (/ (/ l t_m) t_m) (* k (* t_m k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 2.5e-10) {
		tmp = (l * (l / k)) / (t_m * (t_m * (t_m * k)));
	} else {
		tmp = l * (((l / t_m) / 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 (l <= 2.5d-10) then
        tmp = (l * (l / k)) / (t_m * (t_m * (t_m * k)))
    else
        tmp = l * (((l / t_m) / 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 (l <= 2.5e-10) {
		tmp = (l * (l / k)) / (t_m * (t_m * (t_m * k)));
	} else {
		tmp = l * (((l / t_m) / 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 l <= 2.5e-10:
		tmp = (l * (l / k)) / (t_m * (t_m * (t_m * k)))
	else:
		tmp = l * (((l / t_m) / 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 (l <= 2.5e-10)
		tmp = Float64(Float64(l * Float64(l / k)) / Float64(t_m * Float64(t_m * Float64(t_m * k))));
	else
		tmp = Float64(l * Float64(Float64(Float64(l / t_m) / t_m) / Float64(k * Float64(t_m * k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 2.5e-10)
		tmp = (l * (l / k)) / (t_m * (t_m * (t_m * k)));
	else
		tmp = l * (((l / t_m) / 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[l, 2.5e-10], N[(N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(l / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.50000000000000016e-10

   1. Initial program 58.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 54.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 54.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{t}}\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{\ell}{t \cdot t}}{\color{blue}{t}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot t}\right), \color{blue}{t}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot t\right)\right), t\right)\right) \]

      10. *-lowering-*.f64 58.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right), t\right)\right) \]

   7. Applied egg-rr 58.0%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\frac{\ell}{t \cdot t}}{t}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell}{k}}{k} \cdot \frac{\color{blue}{\frac{\ell}{t \cdot t}}}{t} \]

      2. associate-/l/ N/A

         \[\leadsto \frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{\color{blue}{t \cdot \left(t \cdot t\right)}} \]

      3. frac-times N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{\color{blue}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{k}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{k} \cdot \ell\right), \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{k}\right), \ell\right), \left(\color{blue}{t} \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \ell\right), \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \ell\right), \mathsf{*.f64}\left(t, \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \ell\right), \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{\left(t \cdot k\right)}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \ell\right), \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot \color{blue}{t}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \ell\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(k \cdot t\right)}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \ell\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{k}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 64.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \ell\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{k}\right)\right)\right)\right) \]

   9. Applied egg-rr 64.9%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \ell}{t \cdot \left(t \cdot \left(t \cdot k\right)\right)}} \]

   if 2.50000000000000016e-10 < l

   1. Initial program 43.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 44.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 44.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{t}}\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{\ell}{t \cdot t}}{\color{blue}{t}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot t}\right), \color{blue}{t}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot t\right)\right), t\right)\right) \]

      10. *-lowering-*.f64 49.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right), t\right)\right) \]

   7. Applied egg-rr 49.9%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\frac{\ell}{t \cdot t}}{t}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{\ell}{t \cdot t}}{t} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

      2. associate-/l/ N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot t\right)} \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]

      3. frac-times N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot \color{blue}{k}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right) \cdot k} \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \]

      7. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \color{blue}{\left(k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)\right)}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right) \cdot \color{blue}{k}\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot k\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot k\right)}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(\left(t \cdot t\right) \cdot k\right), \color{blue}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot k\right)\right), k\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot \left(k \cdot t\right)\right), k\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot t\right)\right), k\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot k\right)\right), k\right)\right)\right)\right) \]

      18. *-lowering-*.f64 54.7%

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, k\right)\right), k\right)\right)\right)\right) \]

   9. Applied egg-rr 54.7%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot \left(t \cdot k\right)\right) \cdot k\right)}} \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \left(\frac{\frac{\ell}{t}}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right) \cdot k}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \left(\frac{\frac{\ell}{t}}{t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}}\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \left(\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)}\right)\right) \]

       4. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \left(\frac{\frac{\frac{\ell}{t}}{t}}{\color{blue}{t \cdot \left(k \cdot k\right)}}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{t}\right), \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), t\right), \left(\color{blue}{t} \cdot \left(k \cdot k\right)\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), t\right), \left(t \cdot \left(k \cdot k\right)\right)\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), t\right), \left(\left(t \cdot k\right) \cdot \color{blue}{k}\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), t\right), \left(k \cdot \color{blue}{\left(t \cdot k\right)}\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), t\right), \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot k\right)}\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), t\right), \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{t}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 54.7%

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), t\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{t}\right)\right)\right)\right) \]

   11. Applied egg-rr 54.7%

       \[\leadsto \ell \cdot \color{blue}{\frac{\frac{\frac{\ell}{t}}{t}}{k \cdot \left(k \cdot t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{k}}{t \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t}}{t}}{k \cdot \left(t \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.7% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \frac{\frac{\ell}{t\_m \cdot t\_m}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2e-42)
    (* (/ (/ l k) k) (/ (/ l (* t_m t_m)) t_m))
    (* l (/ l (* t_m (* k (* t_m (* t_m k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2e-42) {
		tmp = ((l / k) / k) * ((l / (t_m * t_m)) / t_m);
	} else {
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2d-42) then
        tmp = ((l / k) / k) * ((l / (t_m * t_m)) / t_m)
    else
        tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2e-42) {
		tmp = ((l / k) / k) * ((l / (t_m * t_m)) / t_m);
	} else {
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2e-42:
		tmp = ((l / k) / k) * ((l / (t_m * t_m)) / t_m)
	else:
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2e-42)
		tmp = Float64(Float64(Float64(l / k) / k) * Float64(Float64(l / Float64(t_m * t_m)) / t_m));
	else
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(k * Float64(t_m * Float64(t_m * k))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2e-42)
		tmp = ((l / k) / k) * ((l / (t_m * t_m)) / t_m);
	else
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2e-42], N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(t$95$m * N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.00000000000000008e-42

   1. Initial program 48.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 45.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 45.8%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{t}}\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{\ell}{t \cdot t}}{\color{blue}{t}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot t}\right), \color{blue}{t}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot t\right)\right), t\right)\right) \]

      10. *-lowering-*.f64 50.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right), t\right)\right) \]

   7. Applied egg-rr 50.1%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\frac{\ell}{t \cdot t}}{t}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{k}}{k}\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right)}, t\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{k}\right), k\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right)}, t\right)\right) \]

      3. /-lowering-/.f64 52.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(t, t\right)\right), t\right)\right) \]

   9. Applied egg-rr 52.9%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{k}} \cdot \frac{\frac{\ell}{t \cdot t}}{t} \]

   if 2.00000000000000008e-42 < t

   1. Initial program 67.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 65.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 65.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{t}}\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{\ell}{t \cdot t}}{\color{blue}{t}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot t}\right), \color{blue}{t}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot t\right)\right), t\right)\right) \]

      10. *-lowering-*.f64 69.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right), t\right)\right) \]

   7. Applied egg-rr 69.3%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\frac{\ell}{t \cdot t}}{t}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{\ell}{t \cdot t}}{t} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

      2. associate-/l/ N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot t\right)} \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]

      3. frac-times N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot \color{blue}{k}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right) \cdot k} \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \]

      7. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \color{blue}{\left(k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)\right)}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right) \cdot \color{blue}{k}\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot k\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot k\right)}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(\left(t \cdot t\right) \cdot k\right), \color{blue}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot k\right)\right), k\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot \left(k \cdot t\right)\right), k\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot t\right)\right), k\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot k\right)\right), k\right)\right)\right)\right) \]

      18. *-lowering-*.f64 80.9%

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, k\right)\right), k\right)\right)\right)\right) \]

   9. Applied egg-rr 80.9%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot \left(t \cdot k\right)\right) \cdot k\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \frac{\frac{\ell}{t \cdot t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.8% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-248}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot t\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t\_m}}{t\_m}}{k \cdot \left(t\_m \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.5e-248)
    (* (/ l k) (/ l (* t_m (* k (* t_m t_m)))))
    (* l (/ (/ (/ l t_m) t_m) (* k (* t_m k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.5e-248) {
		tmp = (l / k) * (l / (t_m * (k * (t_m * t_m))));
	} else {
		tmp = l * (((l / t_m) / 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 (k <= 2.5d-248) then
        tmp = (l / k) * (l / (t_m * (k * (t_m * t_m))))
    else
        tmp = l * (((l / t_m) / 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 (k <= 2.5e-248) {
		tmp = (l / k) * (l / (t_m * (k * (t_m * t_m))));
	} else {
		tmp = l * (((l / t_m) / 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 k <= 2.5e-248:
		tmp = (l / k) * (l / (t_m * (k * (t_m * t_m))))
	else:
		tmp = l * (((l / t_m) / 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 (k <= 2.5e-248)
		tmp = Float64(Float64(l / k) * Float64(l / Float64(t_m * Float64(k * Float64(t_m * t_m)))));
	else
		tmp = Float64(l * Float64(Float64(Float64(l / t_m) / t_m) / Float64(k * Float64(t_m * k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 2.5e-248)
		tmp = (l / k) * (l / (t_m * (k * (t_m * t_m))));
	else
		tmp = l * (((l / t_m) / 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[k, 2.5e-248], N[(N[(l / k), $MachinePrecision] * N[(l / N[(t$95$m * N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(l / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.5e-248

   1. Initial program 51.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 46.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 46.8%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right) \cdot \color{blue}{k}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)} \cdot \color{blue}{\frac{\ell}{k}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\right), \color{blue}{\left(\frac{\ell}{k}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\color{blue}{\ell}}{k}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(t \cdot t\right) \cdot k\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot t\right), k\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right), \left(\frac{\ell}{k}\right)\right) \]

      10. /-lowering-/.f64 56.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), k\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{k}\right)\right) \]

   7. Applied egg-rr 56.2%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot k\right)} \cdot \frac{\ell}{k}} \]

   if 2.5e-248 < k

   1. Initial program 56.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 56.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 56.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{t}}\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{\ell}{t \cdot t}}{\color{blue}{t}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot t}\right), \color{blue}{t}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot t\right)\right), t\right)\right) \]

      10. *-lowering-*.f64 63.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right), t\right)\right) \]

   7. Applied egg-rr 63.1%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\frac{\ell}{t \cdot t}}{t}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{\ell}{t \cdot t}}{t} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

      2. associate-/l/ N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot t\right)} \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]

      3. frac-times N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot \color{blue}{k}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right) \cdot k} \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \]

      7. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \color{blue}{\left(k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)\right)}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right) \cdot \color{blue}{k}\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot k\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot k\right)}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(\left(t \cdot t\right) \cdot k\right), \color{blue}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot k\right)\right), k\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot \left(k \cdot t\right)\right), k\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot t\right)\right), k\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot k\right)\right), k\right)\right)\right)\right) \]

      18. *-lowering-*.f64 66.2%

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, k\right)\right), k\right)\right)\right)\right) \]

   9. Applied egg-rr 66.2%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot \left(t \cdot k\right)\right) \cdot k\right)}} \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \left(\frac{\frac{\ell}{t}}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right) \cdot k}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \left(\frac{\frac{\ell}{t}}{t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}}\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \left(\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)}\right)\right) \]

       4. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \left(\frac{\frac{\frac{\ell}{t}}{t}}{\color{blue}{t \cdot \left(k \cdot k\right)}}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{t}\right), \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), t\right), \left(\color{blue}{t} \cdot \left(k \cdot k\right)\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), t\right), \left(t \cdot \left(k \cdot k\right)\right)\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), t\right), \left(\left(t \cdot k\right) \cdot \color{blue}{k}\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), t\right), \left(k \cdot \color{blue}{\left(t \cdot k\right)}\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), t\right), \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot k\right)}\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), t\right), \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{t}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 67.8%

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), t\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{t}\right)\right)\right)\right) \]

   11. Applied egg-rr 67.8%

       \[\leadsto \ell \cdot \color{blue}{\frac{\frac{\frac{\ell}{t}}{t}}{k \cdot \left(k \cdot t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-248}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t}}{t}}{k \cdot \left(t \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 66.6% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-161}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t\_m}}{t\_m}}{k \cdot \left(t\_m \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 4e-161)
    (* l (/ l (* t_m (* k (* t_m (* t_m k))))))
    (* l (/ (/ (/ l t_m) t_m) (* k (* t_m k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 4e-161) {
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))));
	} else {
		tmp = l * (((l / t_m) / 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 (k <= 4d-161) then
        tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))))
    else
        tmp = l * (((l / t_m) / 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 (k <= 4e-161) {
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))));
	} else {
		tmp = l * (((l / t_m) / 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 k <= 4e-161:
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))))
	else:
		tmp = l * (((l / t_m) / 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 (k <= 4e-161)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(k * Float64(t_m * Float64(t_m * k))))));
	else
		tmp = Float64(l * Float64(Float64(Float64(l / t_m) / t_m) / Float64(k * Float64(t_m * k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 4e-161)
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))));
	else
		tmp = l * (((l / t_m) / 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[k, 4e-161], N[(l * N[(l / N[(t$95$m * N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(l / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.00000000000000011e-161

   1. Initial program 53.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 50.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 50.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{t}}\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{\ell}{t \cdot t}}{\color{blue}{t}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot t}\right), \color{blue}{t}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot t\right)\right), t\right)\right) \]

      10. *-lowering-*.f64 50.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right), t\right)\right) \]

   7. Applied egg-rr 50.9%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\frac{\ell}{t \cdot t}}{t}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{\ell}{t \cdot t}}{t} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

      2. associate-/l/ N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot t\right)} \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]

      3. frac-times N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot \color{blue}{k}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right) \cdot k} \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \]

      7. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \color{blue}{\left(k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)\right)}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right) \cdot \color{blue}{k}\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot k\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot k\right)}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(\left(t \cdot t\right) \cdot k\right), \color{blue}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot k\right)\right), k\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot \left(k \cdot t\right)\right), k\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot t\right)\right), k\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot k\right)\right), k\right)\right)\right)\right) \]

      18. *-lowering-*.f64 62.7%

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, k\right)\right), k\right)\right)\right)\right) \]

   9. Applied egg-rr 62.7%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot \left(t \cdot k\right)\right) \cdot k\right)}} \]

   if 4.00000000000000011e-161 < k

   1. Initial program 55.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 53.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 53.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{t}}\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{\ell}{t \cdot t}}{\color{blue}{t}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot t}\right), \color{blue}{t}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot t\right)\right), t\right)\right) \]

      10. *-lowering-*.f64 62.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right), t\right)\right) \]

   7. Applied egg-rr 62.2%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\frac{\ell}{t \cdot t}}{t}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{\ell}{t \cdot t}}{t} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

      2. associate-/l/ N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot t\right)} \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]

      3. frac-times N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot \color{blue}{k}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right) \cdot k} \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \]

      7. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \color{blue}{\left(k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)\right)}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right) \cdot \color{blue}{k}\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot k\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot k\right)}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(\left(t \cdot t\right) \cdot k\right), \color{blue}{k}\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot k\right)\right), k\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot \left(k \cdot t\right)\right), k\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot t\right)\right), k\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot k\right)\right), k\right)\right)\right)\right) \]

      18. *-lowering-*.f64 61.9%

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, k\right)\right), k\right)\right)\right)\right) \]

   9. Applied egg-rr 61.9%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot \left(t \cdot k\right)\right) \cdot k\right)}} \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \left(\frac{\frac{\ell}{t}}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right) \cdot k}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \left(\frac{\frac{\ell}{t}}{t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}}\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \left(\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)}\right)\right) \]

       4. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \left(\frac{\frac{\frac{\ell}{t}}{t}}{\color{blue}{t \cdot \left(k \cdot k\right)}}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{t}\right), \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), t\right), \left(\color{blue}{t} \cdot \left(k \cdot k\right)\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), t\right), \left(t \cdot \left(k \cdot k\right)\right)\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), t\right), \left(\left(t \cdot k\right) \cdot \color{blue}{k}\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), t\right), \left(k \cdot \color{blue}{\left(t \cdot k\right)}\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), t\right), \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot k\right)}\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), t\right), \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{t}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 64.4%

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), t\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{t}\right)\right)\right)\right) \]

   11. Applied egg-rr 64.4%

       \[\leadsto \ell \cdot \color{blue}{\frac{\frac{\frac{\ell}{t}}{t}}{k \cdot \left(k \cdot t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-161}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t}}{t}}{k \cdot \left(t \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 65.8% accurate, 32.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* l (/ l (* t_m (* k (* t_m (* t_m k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (t_m * (k * (t_m * (t_m * k))))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (l * (l / (t_m * (k * (t_m * (t_m * k))))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (t_m * (k * (t_m * (t_m * k))))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * (l / (t_m * (k * (t_m * (t_m * k))))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(l / Float64(t_m * Float64(k * Float64(t_m * Float64(t_m * k)))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (l * (l / (t_m * (k * (t_m * (t_m * k))))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(l * N[(l / N[(t$95$m * N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
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. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

   8. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 51.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

5. Simplified 51.6%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

   2. times-frac N/A

      \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot t\right)}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(t \cdot t\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{t}}\right)\right) \]

   7. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{\ell}{t \cdot t}}{\color{blue}{t}}\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot t}\right), \color{blue}{t}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot t\right)\right), t\right)\right) \]

   10. *-lowering-*.f64 55.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right), t\right)\right) \]

7. Applied egg-rr 55.9%

   \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\frac{\ell}{t \cdot t}}{t}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\frac{\ell}{t \cdot t}}{t} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

   2. associate-/l/ N/A

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot t\right)} \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]

   3. frac-times N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot \color{blue}{k}} \]

   5. associate-*r* N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right) \cdot k} \]

   6. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \]

   7. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}\right)}\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \color{blue}{\left(k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)\right)}\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right) \cdot \color{blue}{k}\right)\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot k\right)}\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot k\right)}\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(\left(t \cdot t\right) \cdot k\right), \color{blue}{k}\right)\right)\right)\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot k\right)\right), k\right)\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot \left(k \cdot t\right)\right), k\right)\right)\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot t\right)\right), k\right)\right)\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot k\right)\right), k\right)\right)\right)\right) \]

   18. *-lowering-*.f64 62.3%

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, k\right)\right), k\right)\right)\right)\right) \]

9. Applied egg-rr 62.3%

   \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot \left(t \cdot k\right)\right) \cdot k\right)}} \]

10. Final simplification 62.3%

    \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)} \]
11. Add Preprocessing

Alternative 17: 64.5% accurate, 32.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot \left(t\_m \cdot k\right)\right)\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* l (/ l (* t_m (* t_m (* k (* t_m k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (t_m * (t_m * (k * (t_m * k))))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (l * (l / (t_m * (t_m * (k * (t_m * k))))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (t_m * (t_m * (k * (t_m * k))))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * (l / (t_m * (t_m * (k * (t_m * k))))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(l / Float64(t_m * Float64(t_m * Float64(k * Float64(t_m * k)))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (l * (l / (t_m * (t_m * (k * (t_m * k))))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(l * N[(l / N[(t$95$m * N[(t$95$m * N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
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. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot {t}^{3}\right)}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left({t}^{3}\right)}\right)\right)\right) \]

   8. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(t \cdot {t}^{\color{blue}{2}}\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \color{blue}{\left({t}^{2}\right)}\right)\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{t}\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 51.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right)\right) \]

5. Simplified 51.6%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

   2. times-frac N/A

      \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot t\right)}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot \left(t \cdot t\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{t \cdot \left(t \cdot t\right)}\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{t}}\right)\right) \]

   7. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\frac{\ell}{t \cdot t}}{\color{blue}{t}}\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot t}\right), \color{blue}{t}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot t\right)\right), t\right)\right) \]

   10. *-lowering-*.f64 55.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, t\right)\right), t\right)\right) \]

7. Applied egg-rr 55.9%

   \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\frac{\ell}{t \cdot t}}{t}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\frac{\ell}{t \cdot t}}{t} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

   2. associate-/l/ N/A

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot t\right)} \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]

   3. frac-times N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot \color{blue}{k}} \]

   5. associate-*r* N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right) \cdot k} \]

   6. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \]

   7. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}\right)}\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \color{blue}{\left(k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)\right)}\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right) \cdot \color{blue}{k}\right)\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot k\right)}\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot k\right)}\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(\left(t \cdot t\right) \cdot k\right), \color{blue}{k}\right)\right)\right)\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot k\right)\right), k\right)\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot \left(k \cdot t\right)\right), k\right)\right)\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot t\right)\right), k\right)\right)\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot k\right)\right), k\right)\right)\right)\right) \]

   18. *-lowering-*.f64 62.3%

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, k\right)\right), k\right)\right)\right)\right) \]

9. Applied egg-rr 62.3%

   \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot \left(t \cdot k\right)\right) \cdot k\right)}} \]
10. Step-by-step derivation

    1. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)\right)\right)\right) \]

    2. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(t \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}\right)\right)\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot \left(k \cdot k\right)\right), \color{blue}{t}\right)\right)\right)\right) \]

    5. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(\left(t \cdot k\right) \cdot k\right), t\right)\right)\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(k \cdot \left(t \cdot k\right)\right), t\right)\right)\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \left(t \cdot k\right)\right), t\right)\right)\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot t\right)\right), t\right)\right)\right)\right) \]

    9. *-lowering-*.f64 61.1%

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, t\right)\right), t\right)\right)\right)\right) \]

11. Applied egg-rr 61.1%

    \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(k \cdot \left(k \cdot t\right)\right) \cdot t\right)}} \]

12. Final simplification 61.1%

    \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(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))))