Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.0% → 92.7%

Time: 18.1s

Alternatives: 14

Speedup: 32.4×

• 26.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.0% 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: 92.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{t\_m \cdot k}\\ t_3 := \frac{\sin k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.2 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{2}{\tan k}}{{t\_2}^{-1} \cdot {\left(\frac{\frac{\ell}{\sin k}}{k}\right)}^{-1}}\\ \mathbf{elif}\;t\_m \leq 2.75 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m}}{\frac{k \cdot t\_3}{\frac{\ell}{k}} + \frac{2 \cdot \left(t\_m \cdot t\_m\right)}{\frac{\ell}{t\_3}}}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{t\_m \cdot k} \cdot t\_2\\ \end{array} \end{array} \end{array} \]

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 k))) (t_3 (/ (sin k) l)))
   (*
    t_s
    (if (<= t_m 5.2e-130)
      (/ (/ 2.0 (tan k)) (* (pow t_2 -1.0) (pow (/ (/ l (sin k)) k) -1.0)))
      (if (<= t_m 2.75e+132)
        (/
         (/
          (/ 2.0 t_m)
          (+ (/ (* k t_3) (/ l k)) (/ (* 2.0 (* t_m t_m)) (/ l t_3))))
         (tan k))
        (* (/ (/ l t_m) (* t_m 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 * k);
	double t_3 = sin(k) / l;
	double tmp;
	if (t_m <= 5.2e-130) {
		tmp = (2.0 / tan(k)) / (pow(t_2, -1.0) * pow(((l / sin(k)) / k), -1.0));
	} else if (t_m <= 2.75e+132) {
		tmp = ((2.0 / t_m) / (((k * t_3) / (l / k)) + ((2.0 * (t_m * t_m)) / (l / t_3)))) / tan(k);
	} else {
		tmp = ((l / t_m) / (t_m * k)) * 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) :: t_3
    real(8) :: tmp
    t_2 = l / (t_m * k)
    t_3 = sin(k) / l
    if (t_m <= 5.2d-130) then
        tmp = (2.0d0 / tan(k)) / ((t_2 ** (-1.0d0)) * (((l / sin(k)) / k) ** (-1.0d0)))
    else if (t_m <= 2.75d+132) then
        tmp = ((2.0d0 / t_m) / (((k * t_3) / (l / k)) + ((2.0d0 * (t_m * t_m)) / (l / t_3)))) / tan(k)
    else
        tmp = ((l / t_m) / (t_m * k)) * 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 * k);
	double t_3 = Math.sin(k) / l;
	double tmp;
	if (t_m <= 5.2e-130) {
		tmp = (2.0 / Math.tan(k)) / (Math.pow(t_2, -1.0) * Math.pow(((l / Math.sin(k)) / k), -1.0));
	} else if (t_m <= 2.75e+132) {
		tmp = ((2.0 / t_m) / (((k * t_3) / (l / k)) + ((2.0 * (t_m * t_m)) / (l / t_3)))) / Math.tan(k);
	} else {
		tmp = ((l / t_m) / (t_m * k)) * 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 * k)
	t_3 = math.sin(k) / l
	tmp = 0
	if t_m <= 5.2e-130:
		tmp = (2.0 / math.tan(k)) / (math.pow(t_2, -1.0) * math.pow(((l / math.sin(k)) / k), -1.0))
	elif t_m <= 2.75e+132:
		tmp = ((2.0 / t_m) / (((k * t_3) / (l / k)) + ((2.0 * (t_m * t_m)) / (l / t_3)))) / math.tan(k)
	else:
		tmp = ((l / t_m) / (t_m * k)) * 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(l / Float64(t_m * k))
	t_3 = Float64(sin(k) / l)
	tmp = 0.0
	if (t_m <= 5.2e-130)
		tmp = Float64(Float64(2.0 / tan(k)) / Float64((t_2 ^ -1.0) * (Float64(Float64(l / sin(k)) / k) ^ -1.0)));
	elseif (t_m <= 2.75e+132)
		tmp = Float64(Float64(Float64(2.0 / t_m) / Float64(Float64(Float64(k * t_3) / Float64(l / k)) + Float64(Float64(2.0 * Float64(t_m * t_m)) / Float64(l / t_3)))) / tan(k));
	else
		tmp = Float64(Float64(Float64(l / t_m) / Float64(t_m * k)) * 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 * k);
	t_3 = sin(k) / l;
	tmp = 0.0;
	if (t_m <= 5.2e-130)
		tmp = (2.0 / tan(k)) / ((t_2 ^ -1.0) * (((l / sin(k)) / k) ^ -1.0));
	elseif (t_m <= 2.75e+132)
		tmp = ((2.0 / t_m) / (((k * t_3) / (l / k)) + ((2.0 * (t_m * t_m)) / (l / t_3)))) / tan(k);
	else
		tmp = ((l / t_m) / (t_m * k)) * 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[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.2e-130], N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$2, -1.0], $MachinePrecision] * N[Power[N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.75e+132], N[(N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(N[(k * t$95$3), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(l / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 5.2000000000000001e-130

   1. Initial program 48.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

   4. Applied egg-rr 58.2%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 63.1%

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

   7. Simplified 63.1%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. frac-times N/A

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

      7. associate-/r* N/A

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

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

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

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

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

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

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

      11. frac-times N/A

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

      12. associate-*l/ N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

   9. Applied egg-rr 63.1%

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

       1. clear-num N/A

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

       2. inv-pow N/A

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

       3. times-frac N/A

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

       4. times-frac N/A

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

       5. associate-/r* N/A

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

       6. unpow-prod-down N/A

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

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

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

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

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

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

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

       10. *-commutative N/A

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

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

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

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

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

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

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

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

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

       15. sin-lowering-sin.f64 79.6%

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

   11. Applied egg-rr 79.6%

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

   if 5.2000000000000001e-130 < t < 2.75e132

   1. Initial program 60.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

   4. Applied egg-rr 72.7%

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

   5. Taylor expanded in t around 0

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

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

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

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

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

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

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

      4. associate-/l* N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. times-frac N/A

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

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

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

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

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

      16. unpow2 N/A

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

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

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

      18. /-lowering-/.f64 N/A

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

      19. sin-lowering-sin.f64 78.8%

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

   7. Simplified 78.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      10. associate-*r* N/A

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

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

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

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

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

      16. associate-/l* N/A

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

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

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

      18. /-lowering-/.f64 N/A

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

      19. sin-lowering-sin.f64 92.6%

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

   9. Applied egg-rr 92.6%

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

       1. associate-/r* N/A

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

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

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

   11. Applied egg-rr 92.7%

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

   if 2.75e132 < t

   1. Initial program 62.3%

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

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

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

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 61.8%

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

   5. Simplified 61.8%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 70.7%

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

   7. Applied egg-rr 70.7%

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

      1. associate-*r/ N/A

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

      2. unswap-sqr N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      9. *-lowering-*.f64 92.6%

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

   9. Applied egg-rr 92.6%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.2 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{2}{\tan k}}{{\left(\frac{\ell}{t \cdot k}\right)}^{-1} \cdot {\left(\frac{\frac{\ell}{\sin k}}{k}\right)}^{-1}}\\ \mathbf{elif}\;t \leq 2.75 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{\frac{2}{t}}{\frac{k \cdot \frac{\sin k}{\ell}}{\frac{\ell}{k}} + \frac{2 \cdot \left(t \cdot t\right)}{\frac{\ell}{\frac{\sin k}{\ell}}}}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{\ell}{t \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k}{\ell}\\ t_3 := k \cdot t\_2\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.08 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{2}{\tan k}}{t\_3 \cdot \frac{k}{\frac{\ell}{t\_m}}}\\ \mathbf{elif}\;t\_m \leq 2.9 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m}}{\frac{t\_3}{\frac{\ell}{k}} + \frac{2 \cdot \left(t\_m \cdot t\_m\right)}{\frac{\ell}{t\_2}}}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{t\_m \cdot k} \cdot \frac{\ell}{t\_m \cdot k}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (sin k) l)) (t_3 (* k t_2)))
   (*
    t_s
    (if (<= t_m 1.08e-145)
      (/ (/ 2.0 (tan k)) (* t_3 (/ k (/ l t_m))))
      (if (<= t_m 2.9e+132)
        (/
         (/ (/ 2.0 t_m) (+ (/ t_3 (/ l k)) (/ (* 2.0 (* t_m t_m)) (/ l t_2))))
         (tan k))
        (* (/ (/ l t_m) (* t_m k)) (/ l (* t_m k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = sin(k) / l;
	double t_3 = k * t_2;
	double tmp;
	if (t_m <= 1.08e-145) {
		tmp = (2.0 / tan(k)) / (t_3 * (k / (l / t_m)));
	} else if (t_m <= 2.9e+132) {
		tmp = ((2.0 / t_m) / ((t_3 / (l / k)) + ((2.0 * (t_m * t_m)) / (l / t_2)))) / tan(k);
	} else {
		tmp = ((l / t_m) / (t_m * k)) * (l / (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) :: t_3
    real(8) :: tmp
    t_2 = sin(k) / l
    t_3 = k * t_2
    if (t_m <= 1.08d-145) then
        tmp = (2.0d0 / tan(k)) / (t_3 * (k / (l / t_m)))
    else if (t_m <= 2.9d+132) then
        tmp = ((2.0d0 / t_m) / ((t_3 / (l / k)) + ((2.0d0 * (t_m * t_m)) / (l / t_2)))) / tan(k)
    else
        tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.sin(k) / l;
	double t_3 = k * t_2;
	double tmp;
	if (t_m <= 1.08e-145) {
		tmp = (2.0 / Math.tan(k)) / (t_3 * (k / (l / t_m)));
	} else if (t_m <= 2.9e+132) {
		tmp = ((2.0 / t_m) / ((t_3 / (l / k)) + ((2.0 * (t_m * t_m)) / (l / t_2)))) / Math.tan(k);
	} else {
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.sin(k) / l
	t_3 = k * t_2
	tmp = 0
	if t_m <= 1.08e-145:
		tmp = (2.0 / math.tan(k)) / (t_3 * (k / (l / t_m)))
	elif t_m <= 2.9e+132:
		tmp = ((2.0 / t_m) / ((t_3 / (l / k)) + ((2.0 * (t_m * t_m)) / (l / t_2)))) / math.tan(k)
	else:
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(sin(k) / l)
	t_3 = Float64(k * t_2)
	tmp = 0.0
	if (t_m <= 1.08e-145)
		tmp = Float64(Float64(2.0 / tan(k)) / Float64(t_3 * Float64(k / Float64(l / t_m))));
	elseif (t_m <= 2.9e+132)
		tmp = Float64(Float64(Float64(2.0 / t_m) / Float64(Float64(t_3 / Float64(l / k)) + Float64(Float64(2.0 * Float64(t_m * t_m)) / Float64(l / t_2)))) / tan(k));
	else
		tmp = Float64(Float64(Float64(l / t_m) / Float64(t_m * k)) * Float64(l / 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)
	t_2 = sin(k) / l;
	t_3 = k * t_2;
	tmp = 0.0;
	if (t_m <= 1.08e-145)
		tmp = (2.0 / tan(k)) / (t_3 * (k / (l / t_m)));
	elseif (t_m <= 2.9e+132)
		tmp = ((2.0 / t_m) / ((t_3 / (l / k)) + ((2.0 * (t_m * t_m)) / (l / t_2)))) / tan(k);
	else
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$3 = N[(k * t$95$2), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.08e-145], N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * N[(k / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.9e+132], N[(N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(t$95$3 / N[(l / k), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(l / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.07999999999999998e-145

   1. Initial program 48.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

   4. Applied egg-rr 57.7%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 62.9%

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

   7. Simplified 62.9%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. frac-times N/A

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

      7. associate-/r* N/A

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

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

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

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

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

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

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

      11. frac-times N/A

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

      12. associate-*l/ N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

   9. Applied egg-rr 62.9%

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

       1. times-frac N/A

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

       2. times-frac N/A

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

       3. un-div-inv N/A

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

       4. clear-num N/A

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

       5. *-commutative N/A

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

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

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

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

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

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

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

       9. *-commutative N/A

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

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

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

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

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

       12. sin-lowering-sin.f64 78.1%

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

   11. Applied egg-rr 78.1%

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

   if 1.07999999999999998e-145 < t < 2.8999999999999999e132

   1. Initial program 58.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

   4. Applied egg-rr 71.8%

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

   5. Taylor expanded in t around 0

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

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

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

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

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

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

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

      4. associate-/l* N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. times-frac N/A

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

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

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

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

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

      16. unpow2 N/A

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

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

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

      18. /-lowering-/.f64 N/A

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

      19. sin-lowering-sin.f64 77.1%

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

   7. Simplified 77.1%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      10. associate-*r* N/A

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

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

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

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

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

      16. associate-/l* N/A

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

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

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

      18. /-lowering-/.f64 N/A

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

      19. sin-lowering-sin.f64 93.5%

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

   9. Applied egg-rr 93.5%

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

       1. associate-/r* N/A

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

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

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

   11. Applied egg-rr 93.7%

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

   if 2.8999999999999999e132 < t

   1. Initial program 62.3%

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

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

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

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 61.8%

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

   5. Simplified 61.8%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 70.7%

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

   7. Applied egg-rr 70.7%

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

      1. associate-*r/ N/A

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

      2. unswap-sqr N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      9. *-lowering-*.f64 92.6%

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

   9. Applied egg-rr 92.6%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.08 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{2}{\tan k}}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \frac{k}{\frac{\ell}{t}}}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{\frac{2}{t}}{\frac{k \cdot \frac{\sin k}{\ell}}{\frac{\ell}{k}} + \frac{2 \cdot \left(t \cdot t\right)}{\frac{\ell}{\frac{\sin k}{\ell}}}}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{\ell}{t \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{\ell \cdot \ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.32 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{t\_m \cdot k} \cdot \frac{\ell}{t\_m \cdot k}\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{\sin k \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right)}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \left(t\_2 + \left(k \cdot k\right) \cdot \left(\frac{0.08611111111111111}{\ell} \cdot \frac{t\_m \cdot \left(k \cdot k\right)}{\ell} + t\_2 \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \end{array} \]

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 (/ t_m (* l l))))
   (*
    t_s
    (if (<= k 1.32e-14)
      (* (/ (/ l t_m) (* t_m k)) (/ l (* t_m k)))
      (if (<= k 1.55e+99)
        (/ (/ (* l l) (* (sin k) (* t_m (* t_m t_m)))) (tan k))
        (/
         2.0
         (*
          (pow k 4.0)
          (+
           t_2
           (*
            (* k k)
            (+
             (* (/ 0.08611111111111111 l) (/ (* t_m (* k k)) l))
             (* t_2 0.16666666666666666)))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m / (l * l);
	double tmp;
	if (k <= 1.32e-14) {
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k));
	} else if (k <= 1.55e+99) {
		tmp = ((l * l) / (sin(k) * (t_m * (t_m * t_m)))) / tan(k);
	} else {
		tmp = 2.0 / (pow(k, 4.0) * (t_2 + ((k * k) * (((0.08611111111111111 / l) * ((t_m * (k * k)) / l)) + (t_2 * 0.16666666666666666)))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m / (l * l)
    if (k <= 1.32d-14) then
        tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k))
    else if (k <= 1.55d+99) then
        tmp = ((l * l) / (sin(k) * (t_m * (t_m * t_m)))) / tan(k)
    else
        tmp = 2.0d0 / ((k ** 4.0d0) * (t_2 + ((k * k) * (((0.08611111111111111d0 / l) * ((t_m * (k * k)) / l)) + (t_2 * 0.16666666666666666d0)))))
    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 = t_m / (l * l);
	double tmp;
	if (k <= 1.32e-14) {
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k));
	} else if (k <= 1.55e+99) {
		tmp = ((l * l) / (Math.sin(k) * (t_m * (t_m * t_m)))) / Math.tan(k);
	} else {
		tmp = 2.0 / (Math.pow(k, 4.0) * (t_2 + ((k * k) * (((0.08611111111111111 / l) * ((t_m * (k * k)) / l)) + (t_2 * 0.16666666666666666)))));
	}
	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 = t_m / (l * l)
	tmp = 0
	if k <= 1.32e-14:
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k))
	elif k <= 1.55e+99:
		tmp = ((l * l) / (math.sin(k) * (t_m * (t_m * t_m)))) / math.tan(k)
	else:
		tmp = 2.0 / (math.pow(k, 4.0) * (t_2 + ((k * k) * (((0.08611111111111111 / l) * ((t_m * (k * k)) / l)) + (t_2 * 0.16666666666666666)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m / Float64(l * l))
	tmp = 0.0
	if (k <= 1.32e-14)
		tmp = Float64(Float64(Float64(l / t_m) / Float64(t_m * k)) * Float64(l / Float64(t_m * k)));
	elseif (k <= 1.55e+99)
		tmp = Float64(Float64(Float64(l * l) / Float64(sin(k) * Float64(t_m * Float64(t_m * t_m)))) / tan(k));
	else
		tmp = Float64(2.0 / Float64((k ^ 4.0) * Float64(t_2 + Float64(Float64(k * k) * Float64(Float64(Float64(0.08611111111111111 / l) * Float64(Float64(t_m * Float64(k * k)) / l)) + Float64(t_2 * 0.16666666666666666))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = t_m / (l * l);
	tmp = 0.0;
	if (k <= 1.32e-14)
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k));
	elseif (k <= 1.55e+99)
		tmp = ((l * l) / (sin(k) * (t_m * (t_m * t_m)))) / tan(k);
	else
		tmp = 2.0 / ((k ^ 4.0) * (t_2 + ((k * k) * (((0.08611111111111111 / l) * ((t_m * (k * k)) / l)) + (t_2 * 0.16666666666666666)))));
	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[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1.32e-14], N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.55e+99], N[(N[(N[(l * l), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] * N[(t$95$2 + N[(N[(k * k), $MachinePrecision] * N[(N[(N[(0.08611111111111111 / l), $MachinePrecision] * N[(N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 1.32e-14

   1. Initial program 53.7%

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

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

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

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 48.2%

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

   5. Simplified 48.2%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 58.9%

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

   7. Applied egg-rr 58.9%

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

      1. associate-*r/ N/A

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

      2. unswap-sqr N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      9. *-lowering-*.f64 76.0%

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

   9. Applied egg-rr 76.0%

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

   if 1.32e-14 < k < 1.55e99

   1. Initial program 54.4%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. log-lowering-log.f64 0.0%

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

   4. Applied egg-rr 0.0%

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

   6. Applied egg-rr 59.5%

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

   7. Taylor expanded in t around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 52.2%

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

   9. Simplified 52.2%

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

   if 1.55e99 < k

   1. Initial program 49.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

   4. Applied egg-rr 55.2%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 69.6%

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

   7. Simplified 69.6%

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

   8. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({k}^{4} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{t}{{\ell}^{2}}\right) + \frac{t}{{\ell}^{2}}\right)\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{t}{{\ell}^{2}}\right) + \frac{t}{{\ell}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, 4\right), \left(\color{blue}{{k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{t}{{\ell}^{2}}\right)} + \frac{t}{{\ell}^{2}}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, 4\right), \left(\frac{t}{{\ell}^{2}} + \color{blue}{{k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{t}{{\ell}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, 4\right), \mathsf{+.f64}\left(\left(\frac{t}{{\ell}^{2}}\right), \color{blue}{\left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{t}{{\ell}^{2}}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, 4\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(t, \left({\ell}^{2}\right)\right), \left(\color{blue}{{k}^{2}} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{t}{{\ell}^{2}}\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, 4\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(t, \left(\ell \cdot \ell\right)\right), \left({k}^{\color{blue}{2}} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{t}{{\ell}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, 4\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({k}^{\color{blue}{2}} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{t}{{\ell}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, 4\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{\left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}} + \frac{1}{6} \cdot \frac{t}{{\ell}^{2}}\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, 4\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(\left(k \cdot k\right), \left(\color{blue}{\frac{31}{360} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}}} + \frac{1}{6} \cdot \frac{t}{{\ell}^{2}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, 4\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\color{blue}{\frac{31}{360} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}}} + \frac{1}{6} \cdot \frac{t}{{\ell}^{2}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, 4\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}}\right), \color{blue}{\left(\frac{1}{6} \cdot \frac{t}{{\ell}^{2}}\right)}\right)\right)\right)\right)\right) \]

   10. Simplified 64.5%

       \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \left(\frac{t}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{0.08611111111111111}{\ell} \cdot \frac{\left(k \cdot k\right) \cdot t}{\ell} + 0.16666666666666666 \cdot \frac{t}{\ell \cdot \ell}\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.32 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{\ell}{t \cdot k}\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{\sin k \cdot \left(t \cdot \left(t \cdot t\right)\right)}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \left(\frac{t}{\ell \cdot \ell} + \left(k \cdot k\right) \cdot \left(\frac{0.08611111111111111}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell} + \frac{t}{\ell \cdot \ell} \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.00185:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{t\_m \cdot k} \cdot \frac{\ell}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k}}{\frac{k}{\ell} \cdot \frac{k}{\frac{\ell}{t\_m \cdot \sin k}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 0.00185)
    (* (/ (/ l t_m) (* t_m k)) (/ l (* t_m k)))
    (/ (/ 2.0 (tan k)) (* (/ k l) (/ k (/ l (* t_m (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 <= 0.00185) {
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k));
	} else {
		tmp = (2.0 / tan(k)) / ((k / l) * (k / (l / (t_m * 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 <= 0.00185d0) then
        tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k))
    else
        tmp = (2.0d0 / tan(k)) / ((k / l) * (k / (l / (t_m * 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 <= 0.00185) {
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k));
	} else {
		tmp = (2.0 / Math.tan(k)) / ((k / l) * (k / (l / (t_m * 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 <= 0.00185:
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k))
	else:
		tmp = (2.0 / math.tan(k)) / ((k / l) * (k / (l / (t_m * 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 <= 0.00185)
		tmp = Float64(Float64(Float64(l / t_m) / Float64(t_m * k)) * Float64(l / Float64(t_m * k)));
	else
		tmp = Float64(Float64(2.0 / tan(k)) / Float64(Float64(k / l) * Float64(k / Float64(l / Float64(t_m * 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 <= 0.00185)
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k));
	else
		tmp = (2.0 / tan(k)) / ((k / l) * (k / (l / (t_m * 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, 0.00185], N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[(k / l), $MachinePrecision] * N[(k / N[(l / N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.0018500000000000001

   1. Initial program 54.0%

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

   3. Taylor expanded in k around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 48.4%

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

   5. Simplified 48.4%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 59.1%

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

   7. Applied egg-rr 59.1%

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

      1. associate-*r/ N/A

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

      2. unswap-sqr N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      9. *-lowering-*.f64 76.2%

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

   9. Applied egg-rr 76.2%

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

   if 0.0018500000000000001 < k

   1. Initial program 49.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

   4. Applied egg-rr 56.9%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 69.8%

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

   7. Simplified 69.8%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. frac-times N/A

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

      7. associate-/r* N/A

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

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

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

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

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

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

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

      11. frac-times N/A

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

      12. associate-*l/ N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

   9. Applied egg-rr 69.8%

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

       1. associate-/l* N/A

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

       2. times-frac N/A

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

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

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

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

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

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

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

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

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

       7. *-commutative N/A

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

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

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

       9. sin-lowering-sin.f64 85.9%

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

   11. Applied egg-rr 85.9%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00185:\\ \;\;\;\;\frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{\ell}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k}}{\frac{k}{\ell} \cdot \frac{k}{\frac{\ell}{t \cdot \sin k}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{t\_m \cdot k} \cdot \frac{\ell}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(k \cdot \left(\left(t\_m \cdot k\right) \cdot \frac{\sin k}{\ell \cdot \ell}\right)\right)}\\ \end{array} \end{array} \]

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 (<= k 6.8e-8)
    (* (/ (/ l t_m) (* t_m k)) (/ l (* t_m k)))
    (/ 2.0 (* (tan k) (* k (* (* t_m k) (/ (sin k) (* l l)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.8e-8) {
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k));
	} else {
		tmp = 2.0 / (tan(k) * (k * ((t_m * k) * (sin(k) / (l * l)))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6.8d-8) then
        tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k))
    else
        tmp = 2.0d0 / (tan(k) * (k * ((t_m * k) * (sin(k) / (l * l)))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.8e-8) {
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k));
	} else {
		tmp = 2.0 / (Math.tan(k) * (k * ((t_m * k) * (Math.sin(k) / (l * l)))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 6.8e-8:
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k))
	else:
		tmp = 2.0 / (math.tan(k) * (k * ((t_m * k) * (math.sin(k) / (l * l)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6.8e-8)
		tmp = Float64(Float64(Float64(l / t_m) / Float64(t_m * k)) * Float64(l / Float64(t_m * k)));
	else
		tmp = Float64(2.0 / Float64(tan(k) * Float64(k * Float64(Float64(t_m * k) * Float64(sin(k) / Float64(l * l))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 6.8e-8)
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k));
	else
		tmp = 2.0 / (tan(k) * (k * ((t_m * k) * (sin(k) / (l * l)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6.8e-8], N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(k * N[(N[(t$95$m * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.8e-8

   1. Initial program 54.0%

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

   3. Taylor expanded in k around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 48.4%

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

   5. Simplified 48.4%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 59.1%

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

   7. Applied egg-rr 59.1%

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

      1. associate-*r/ N/A

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

      2. unswap-sqr N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      9. *-lowering-*.f64 76.2%

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

   9. Applied egg-rr 76.2%

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

   if 6.8e-8 < k

   1. Initial program 49.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

   4. Applied egg-rr 56.9%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 69.8%

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

   7. Simplified 69.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 82.8%

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

   9. Applied egg-rr 82.8%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{\ell}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(k \cdot \left(\left(t \cdot k\right) \cdot \frac{\sin k}{\ell \cdot \ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.0013:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{t\_m \cdot k} \cdot \frac{\ell}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \frac{\tan k \cdot \left(t\_m \cdot \sin k\right)}{\ell \cdot \ell}\right)}\\ \end{array} \end{array} \]

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 0.0013)
    (* (/ (/ l t_m) (* t_m k)) (/ l (* t_m k)))
    (/ 2.0 (* k (* k (/ (* (tan k) (* t_m (sin k))) (* l l))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.0013) {
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k));
	} else {
		tmp = 2.0 / (k * (k * ((tan(k) * (t_m * sin(k))) / (l * l))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 0.0013d0) then
        tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k))
    else
        tmp = 2.0d0 / (k * (k * ((tan(k) * (t_m * sin(k))) / (l * l))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.0013) {
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k));
	} else {
		tmp = 2.0 / (k * (k * ((Math.tan(k) * (t_m * Math.sin(k))) / (l * l))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 0.0013:
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k))
	else:
		tmp = 2.0 / (k * (k * ((math.tan(k) * (t_m * math.sin(k))) / (l * l))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 0.0013)
		tmp = Float64(Float64(Float64(l / t_m) / Float64(t_m * k)) * Float64(l / Float64(t_m * k)));
	else
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(tan(k) * Float64(t_m * sin(k))) / Float64(l * l)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 0.0013)
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k));
	else
		tmp = 2.0 / (k * (k * ((tan(k) * (t_m * sin(k))) / (l * l))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 0.0013], N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(k * N[(N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.0012999999999999999

   1. Initial program 54.0%

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

   3. Taylor expanded in k around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 48.4%

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

   5. Simplified 48.4%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 59.1%

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

   7. Applied egg-rr 59.1%

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

      1. associate-*r/ N/A

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

      2. unswap-sqr N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      9. *-lowering-*.f64 76.2%

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

   9. Applied egg-rr 76.2%

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

   if 0.0012999999999999999 < k

   1. Initial program 49.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

   4. Applied egg-rr 56.9%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 69.8%

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

   7. Simplified 69.8%

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

      1. associate-*l* N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. associate-*l/ N/A

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

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

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 73.1%

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

   9. Applied egg-rr 73.1%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0013:\\ \;\;\;\;\frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{\ell}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \frac{\tan k \cdot \left(t \cdot \sin k\right)}{\ell \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.25 \cdot 10^{+40}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{\sin k}}{\frac{t\_m}{\ell}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{t\_m \cdot k} \cdot \frac{\ell}{t\_m \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.25e+40)
    (* (/ 2.0 (tan k)) (/ (/ (/ l (sin k)) (/ t_m l)) (* k k)))
    (* (/ (/ l t_m) (* t_m k)) (/ l (* t_m k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.25e+40) {
		tmp = (2.0 / tan(k)) * (((l / sin(k)) / (t_m / l)) / (k * k));
	} else {
		tmp = ((l / t_m) / (t_m * k)) * (l / (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 <= 1.25d+40) then
        tmp = (2.0d0 / tan(k)) * (((l / sin(k)) / (t_m / l)) / (k * k))
    else
        tmp = ((l / t_m) / (t_m * k)) * (l / (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 <= 1.25e+40) {
		tmp = (2.0 / Math.tan(k)) * (((l / Math.sin(k)) / (t_m / l)) / (k * k));
	} else {
		tmp = ((l / t_m) / (t_m * k)) * (l / (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 <= 1.25e+40:
		tmp = (2.0 / math.tan(k)) * (((l / math.sin(k)) / (t_m / l)) / (k * k))
	else:
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.25e+40)
		tmp = Float64(Float64(2.0 / tan(k)) * Float64(Float64(Float64(l / sin(k)) / Float64(t_m / l)) / Float64(k * k)));
	else
		tmp = Float64(Float64(Float64(l / t_m) / Float64(t_m * k)) * Float64(l / 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 <= 1.25e+40)
		tmp = (2.0 / tan(k)) * (((l / sin(k)) / (t_m / l)) / (k * k));
	else
		tmp = ((l / t_m) / (t_m * k)) * (l / (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, 1.25e+40], N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.25000000000000001e40

   1. Initial program 50.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

   4. Applied egg-rr 59.4%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 64.0%

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

   7. Simplified 64.0%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. frac-times N/A

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

      7. associate-/r* N/A

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

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

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

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

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

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

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

      11. frac-times N/A

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

      12. associate-*l/ N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

   9. Applied egg-rr 63.9%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

          \[\leadsto \frac{1}{\frac{k \cdot k}{\frac{\ell \cdot \ell}{t \cdot \sin k}}} \cdot \color{blue}{\frac{2}{\tan k}} \]

       3. clear-num N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t \cdot \sin k}}{k \cdot k} \cdot \frac{\color{blue}{2}}{\tan k} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell \cdot \ell}{t \cdot \sin k}}{k \cdot k}\right), \color{blue}{\left(\frac{2}{\tan k}\right)}\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell \cdot \ell}{t \cdot \sin k}\right), \left(k \cdot k\right)\right), \left(\frac{\color{blue}{2}}{\tan k}\right)\right) \]

       6. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t} \cdot \frac{\ell}{\sin k}\right), \left(k \cdot k\right)\right), \left(\frac{2}{\tan k}\right)\right) \]

       7. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\frac{t}{\ell}} \cdot \frac{\ell}{\sin k}\right), \left(k \cdot k\right)\right), \left(\frac{2}{\tan k}\right)\right) \]

       8. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \frac{\ell}{\sin k}}{\frac{t}{\ell}}\right), \left(k \cdot k\right)\right), \left(\frac{2}{\tan k}\right)\right) \]

       9. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 \cdot \frac{1}{\frac{\sin k}{\ell}}}{\frac{t}{\ell}}\right), \left(k \cdot k\right)\right), \left(\frac{2}{\tan k}\right)\right) \]

       10. div-inv N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{1}{\frac{\sin k}{\ell}}}{\frac{t}{\ell}}\right), \left(k \cdot k\right)\right), \left(\frac{2}{\tan k}\right)\right) \]

       11. clear-num N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\ell}{\sin k}}{\frac{t}{\ell}}\right), \left(k \cdot k\right)\right), \left(\frac{2}{\tan k}\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{\sin k}\right), \left(\frac{t}{\ell}\right)\right), \left(k \cdot k\right)\right), \left(\frac{2}{\tan k}\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \sin k\right), \left(\frac{t}{\ell}\right)\right), \left(k \cdot k\right)\right), \left(\frac{2}{\tan k}\right)\right) \]

       14. sin-lowering-sin.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{sin.f64}\left(k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(k \cdot k\right)\right), \left(\frac{2}{\tan k}\right)\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(k \cdot k\right)\right), \left(\frac{2}{\tan k}\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{2}{\tan k}\right)\right) \]

       17. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(2, \color{blue}{\tan k}\right)\right) \]

       18. tan-lowering-tan.f64 70.1%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right)\right) \]

   11. Applied egg-rr 70.1%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{\sin k}}{\frac{t}{\ell}}}{k \cdot k} \cdot \frac{2}{\tan k}} \]

   if 1.25000000000000001e40 < t

   1. Initial program 61.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      12. *-lowering-*.f64 57.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   5. Simplified 57.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot t\right), \color{blue}{\left(k \cdot k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{k} \cdot k\right)\right)\right)\right) \]

      8. *-lowering-*.f64 70.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right) \]

   7. Applied egg-rr 70.9%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      2. unswap-sqr N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\left(t \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

      3. times-frac N/A

         \[\leadsto \frac{\frac{\ell}{t}}{t \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot k}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{t \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot k}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(t \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot k}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(t \cdot k\right)\right), \left(\frac{\ell}{t \cdot k}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, k\right)\right), \left(\frac{\ell}{t \cdot k}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot k\right)}\right)\right) \]

      9. *-lowering-*.f64 89.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{k}\right)\right)\right) \]

   9. Applied egg-rr 89.3%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{\ell}{t \cdot k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{+40}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \frac{\frac{\frac{\ell}{\sin k}}{\frac{t}{\ell}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{\ell}{t \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.6% accurate, 13.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{t\_m \cdot k} \cdot \frac{\ell}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\frac{2}{\ell} \cdot \frac{t\_m \cdot \left(t\_m \cdot t\_m\right)}{\ell} + \frac{t\_m \cdot \left(k \cdot k\right)}{\ell \cdot \ell}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 4.5e-43)
    (* (/ (/ l t_m) (* t_m k)) (/ l (* t_m k)))
    (/
     2.0
     (*
      (* k k)
      (+
       (* (/ 2.0 l) (/ (* t_m (* t_m t_m)) l))
       (/ (* t_m (* k k)) (* l l))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 4.5e-43) {
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k));
	} else {
		tmp = 2.0 / ((k * k) * (((2.0 / l) * ((t_m * (t_m * t_m)) / l)) + ((t_m * (k * k)) / (l * l))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4.5d-43) then
        tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k))
    else
        tmp = 2.0d0 / ((k * k) * (((2.0d0 / l) * ((t_m * (t_m * t_m)) / l)) + ((t_m * (k * k)) / (l * l))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 4.5e-43) {
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k));
	} else {
		tmp = 2.0 / ((k * k) * (((2.0 / l) * ((t_m * (t_m * t_m)) / l)) + ((t_m * (k * k)) / (l * l))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 4.5e-43:
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k))
	else:
		tmp = 2.0 / ((k * k) * (((2.0 / l) * ((t_m * (t_m * t_m)) / l)) + ((t_m * (k * k)) / (l * l))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 4.5e-43)
		tmp = Float64(Float64(Float64(l / t_m) / Float64(t_m * k)) * Float64(l / Float64(t_m * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(Float64(2.0 / l) * Float64(Float64(t_m * Float64(t_m * t_m)) / l)) + Float64(Float64(t_m * Float64(k * k)) / Float64(l * l)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 4.5e-43)
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k));
	else
		tmp = 2.0 / ((k * k) * (((2.0 / l) * ((t_m * (t_m * t_m)) / l)) + ((t_m * (k * k)) / (l * l))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 4.5e-43], N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(N[(2.0 / l), $MachinePrecision] * N[(N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.50000000000000025e-43

   1. Initial program 53.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      12. *-lowering-*.f64 47.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   5. Simplified 47.5%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot t\right), \color{blue}{\left(k \cdot k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{k} \cdot k\right)\right)\right)\right) \]

      8. *-lowering-*.f64 58.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right) \]

   7. Applied egg-rr 58.5%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      2. unswap-sqr N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\left(t \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

      3. times-frac N/A

         \[\leadsto \frac{\frac{\ell}{t}}{t \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot k}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{t \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot k}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(t \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot k}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(t \cdot k\right)\right), \left(\frac{\ell}{t \cdot k}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, k\right)\right), \left(\frac{\ell}{t \cdot k}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot k\right)}\right)\right) \]

      9. *-lowering-*.f64 76.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{k}\right)\right)\right) \]

   9. Applied egg-rr 76.0%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{\ell}{t \cdot k}} \]

   if 4.50000000000000025e-43 < k

   1. Initial program 50.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\tan k}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \color{blue}{\tan k}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right), \color{blue}{\tan k}\right)\right) \]

   4. Applied egg-rr 57.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{\frac{t \cdot t}{\ell}}{\ell}\right) \cdot \left(2 + \frac{k}{\frac{t}{\frac{k}{t}}}\right)\right) \cdot \tan k}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)}\right) \]

   7. Simplified 44.6%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \left(\frac{2}{\ell} \cdot \frac{t \cdot \left(t \cdot t\right)}{\ell} + \frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell \cdot \ell} \cdot \left(\frac{1}{t \cdot t} + 0.3333333333333333\right)\right)}} \]

   8. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right)\right), \color{blue}{\left(\frac{{k}^{2} \cdot t}{{\ell}^{2}}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right)\right), \mathsf{/.f64}\left(\left({k}^{2} \cdot t\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), t\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), t\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \left({\ell}^{2}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 59.8%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right)\right) \]

   10. Simplified 59.8%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\frac{2}{\ell} \cdot \frac{t \cdot \left(t \cdot t\right)}{\ell} + \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{\ell}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\frac{2}{\ell} \cdot \frac{t \cdot \left(t \cdot t\right)}{\ell} + \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.2% accurate, 21.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{+19}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{t\_m \cdot k} \cdot \frac{\ell}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t\_m \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.1e+19)
    (* (/ (/ l t_m) (* t_m k)) (/ l (* t_m k)))
    (/ 2.0 (* (* k k) (/ (* t_m (* k k)) (* l l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.1e+19) {
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k));
	} else {
		tmp = 2.0 / ((k * k) * ((t_m * (k * k)) / (l * l)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.1d+19) then
        tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k))
    else
        tmp = 2.0d0 / ((k * k) * ((t_m * (k * k)) / (l * l)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.1e+19) {
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k));
	} else {
		tmp = 2.0 / ((k * k) * ((t_m * (k * k)) / (l * l)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.1e+19:
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k))
	else:
		tmp = 2.0 / ((k * k) * ((t_m * (k * k)) / (l * l)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.1e+19)
		tmp = Float64(Float64(Float64(l / t_m) / Float64(t_m * k)) * Float64(l / Float64(t_m * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(t_m * Float64(k * k)) / Float64(l * l))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.1e+19)
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k));
	else
		tmp = 2.0 / ((k * k) * ((t_m * (k * k)) / (l * l)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.1e+19], N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.1e19

   1. Initial program 54.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      12. *-lowering-*.f64 48.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   5. Simplified 48.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot t\right), \color{blue}{\left(k \cdot k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{k} \cdot k\right)\right)\right)\right) \]

      8. *-lowering-*.f64 58.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right) \]

   7. Applied egg-rr 58.8%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      2. unswap-sqr N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\left(t \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

      3. times-frac N/A

         \[\leadsto \frac{\frac{\ell}{t}}{t \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot k}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{t \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot k}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(t \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot k}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(t \cdot k\right)\right), \left(\frac{\ell}{t \cdot k}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, k\right)\right), \left(\frac{\ell}{t \cdot k}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot k\right)}\right)\right) \]

      9. *-lowering-*.f64 75.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{k}\right)\right)\right) \]

   9. Applied egg-rr 75.7%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{\ell}{t \cdot k}} \]

   if 1.1e19 < k

   1. Initial program 49.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\tan k}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \color{blue}{\tan k}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right), \color{blue}{\tan k}\right)\right) \]

   4. Applied egg-rr 56.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{\frac{t \cdot t}{\ell}}{\ell}\right) \cdot \left(2 + \frac{k}{\frac{t}{\frac{k}{t}}}\right)\right) \cdot \tan k}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)}\right) \]

   7. Simplified 44.1%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \left(\frac{2}{\ell} \cdot \frac{t \cdot \left(t \cdot t\right)}{\ell} + \frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell \cdot \ell} \cdot \left(\frac{1}{t \cdot t} + 0.3333333333333333\right)\right)}} \]

   8. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \color{blue}{\left(\frac{{k}^{2} \cdot t}{{\ell}^{2}}\right)}\right)\right) \]
   9. 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 57.8%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   10. Simplified 57.8%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{+19}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{\ell}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.1% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-97}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{t\_m \cdot k} \cdot \frac{\ell}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot \left(k \cdot k\right)}}{\frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 5e-97)
    (* (/ (/ l t_m) (* t_m k)) (/ l (* t_m k)))
    (/ (/ (/ l t_m) (* t_m (* k k))) (/ t_m l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5e-97) {
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k));
	} else {
		tmp = ((l / t_m) / (t_m * (k * k))) / (t_m / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5d-97) then
        tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k))
    else
        tmp = ((l / t_m) / (t_m * (k * k))) / (t_m / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5e-97) {
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k));
	} else {
		tmp = ((l / t_m) / (t_m * (k * k))) / (t_m / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 5e-97:
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k))
	else:
		tmp = ((l / t_m) / (t_m * (k * k))) / (t_m / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 5e-97)
		tmp = Float64(Float64(Float64(l / t_m) / Float64(t_m * k)) * Float64(l / Float64(t_m * k)));
	else
		tmp = Float64(Float64(Float64(l / t_m) / Float64(t_m * Float64(k * k))) / Float64(t_m / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 5e-97)
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k));
	else
		tmp = ((l / t_m) / (t_m * (k * k))) / (t_m / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 5e-97], N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.9999999999999995e-97

   1. Initial program 53.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      12. *-lowering-*.f64 45.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   5. Simplified 45.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot t\right), \color{blue}{\left(k \cdot k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{k} \cdot k\right)\right)\right)\right) \]

      8. *-lowering-*.f64 57.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right) \]

   7. Applied egg-rr 57.3%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      2. unswap-sqr N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\left(t \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

      3. times-frac N/A

         \[\leadsto \frac{\frac{\ell}{t}}{t \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot k}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{t \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot k}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(t \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot k}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(t \cdot k\right)\right), \left(\frac{\ell}{t \cdot k}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, k\right)\right), \left(\frac{\ell}{t \cdot k}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot k\right)}\right)\right) \]

      9. *-lowering-*.f64 75.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{k}\right)\right)\right) \]

   9. Applied egg-rr 75.5%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{\ell}{t \cdot k}} \]

   if 4.9999999999999995e-97 < k

   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. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      12. *-lowering-*.f64 52.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   5. Simplified 52.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot t\right), \color{blue}{\left(k \cdot k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{k} \cdot k\right)\right)\right)\right) \]

      8. *-lowering-*.f64 59.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right) \]

   7. Applied egg-rr 59.1%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

      2. clear-num N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \frac{1}{\color{blue}{\frac{t}{\ell}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}}{\color{blue}{\frac{t}{\ell}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right), \color{blue}{\left(\frac{t}{\ell}\right)}\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\right), \left(\frac{t}{\ell}\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot k\right)}\right), \left(\frac{\color{blue}{t}}{\ell}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\left(k \cdot k\right) \cdot t}\right), \left(\frac{t}{\ell}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(\left(k \cdot k\right) \cdot t\right)\right), \left(\frac{\color{blue}{t}}{\ell}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\left(k \cdot k\right) \cdot t\right)\right), \left(\frac{t}{\ell}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(t \cdot \left(k \cdot k\right)\right)\right), \left(\frac{t}{\ell}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right), \left(\frac{t}{\ell}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right), \left(\frac{t}{\ell}\right)\right) \]

      13. /-lowering-/.f64 62.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\ell}\right)\right) \]

   9. Applied egg-rr 62.9%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 70.9% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.42 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{t\_m \cdot k} \cdot \frac{\ell}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.42e+20)
    (* (/ (/ l t_m) (* t_m k)) (/ l (* t_m k)))
    (/ (/ (* l l) (* t_m (* t_m (* k k)))) t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.42e+20) {
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k));
	} else {
		tmp = ((l * l) / (t_m * (t_m * (k * k)))) / 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 <= 1.42d+20) then
        tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k))
    else
        tmp = ((l * l) / (t_m * (t_m * (k * k)))) / 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 <= 1.42e+20) {
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k));
	} else {
		tmp = ((l * l) / (t_m * (t_m * (k * k)))) / 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 <= 1.42e+20:
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k))
	else:
		tmp = ((l * l) / (t_m * (t_m * (k * k)))) / 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 <= 1.42e+20)
		tmp = Float64(Float64(Float64(l / t_m) / Float64(t_m * k)) * Float64(l / Float64(t_m * k)));
	else
		tmp = Float64(Float64(Float64(l * l) / Float64(t_m * Float64(t_m * Float64(k * k)))) / 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 <= 1.42e+20)
		tmp = ((l / t_m) / (t_m * k)) * (l / (t_m * k));
	else
		tmp = ((l * l) / (t_m * (t_m * (k * k)))) / 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, 1.42e+20], N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.42e20

   1. Initial program 54.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      12. *-lowering-*.f64 48.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   5. Simplified 48.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot t\right), \color{blue}{\left(k \cdot k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{k} \cdot k\right)\right)\right)\right) \]

      8. *-lowering-*.f64 58.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right) \]

   7. Applied egg-rr 58.8%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      2. unswap-sqr N/A

         \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\left(t \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

      3. times-frac N/A

         \[\leadsto \frac{\frac{\ell}{t}}{t \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot k}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{t \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot k}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(t \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot k}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(t \cdot k\right)\right), \left(\frac{\ell}{t \cdot k}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, k\right)\right), \left(\frac{\ell}{t \cdot k}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot k\right)}\right)\right) \]

      9. *-lowering-*.f64 75.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{k}\right)\right)\right) \]

   9. Applied egg-rr 75.7%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{\ell}{t \cdot k}} \]

   if 1.42e20 < k

   1. Initial program 49.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      12. *-lowering-*.f64 47.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   5. Simplified 47.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot t\right), \color{blue}{\left(k \cdot k\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{k} \cdot k\right)\right)\right)\right) \]

      8. *-lowering-*.f64 55.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right) \]

   7. Applied egg-rr 55.6%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]
   8. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}}{\color{blue}{t}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right), \color{blue}{t}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right), t\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), t\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)\right), t\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), t\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), t\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot t\right)\right)\right), t\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), t\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), t\right) \]

      11. *-lowering-*.f64 57.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), t\right) \]

   9. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 69.8% accurate, 32.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\frac{\ell}{t\_m}}{t\_m \cdot k} \cdot \frac{\ell}{t\_m \cdot k}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (/ (/ l t_m) (* t_m k)) (/ l (* t_m k)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (((l / t_m) / (t_m * k)) * (l / (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 / t_m) / (t_m * k)) * (l / (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 / t_m) / (t_m * k)) * (l / (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 / t_m) / (t_m * k)) * (l / (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(Float64(Float64(l / t_m) / Float64(t_m * k)) * Float64(l / 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 / t_m) / (t_m * k)) * (l / (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[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

   6. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

   12. *-lowering-*.f64 47.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

5. Simplified 47.9%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

   2. times-frac N/A

      \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot t\right), \color{blue}{\left(k \cdot k\right)}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{k} \cdot k\right)\right)\right)\right) \]

   8. *-lowering-*.f64 57.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right) \]

7. Applied egg-rr 57.9%

   \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]
8. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

   2. unswap-sqr N/A

      \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\left(t \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

   3. times-frac N/A

      \[\leadsto \frac{\frac{\ell}{t}}{t \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot k}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t}}{t \cdot k}\right), \color{blue}{\left(\frac{\ell}{t \cdot k}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(t \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{t \cdot k}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(t \cdot k\right)\right), \left(\frac{\ell}{t \cdot k}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, k\right)\right), \left(\frac{\ell}{t \cdot k}\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(t \cdot k\right)}\right)\right) \]

   9. *-lowering-*.f64 69.0%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{k}\right)\right)\right) \]

9. Applied egg-rr 69.0%

   \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{t \cdot k} \cdot \frac{\ell}{t \cdot k}} \]
10. Add Preprocessing

Alternative 13: 67.2% accurate, 32.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\ell}{t\_m} \cdot \frac{\frac{\ell}{t\_m \cdot \left(t\_m \cdot k\right)}}{k}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (/ l t_m) (/ (/ l (* t_m (* t_m k))) k))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / t_m) * ((l / (t_m * (t_m * k))) / k));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l / t_m) * ((l / (t_m * (t_m * k))) / k))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / t_m) * ((l / (t_m * (t_m * k))) / k));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l / t_m) * ((l / (t_m * (t_m * k))) / k))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l / t_m) * Float64(Float64(l / Float64(t_m * Float64(t_m * k))) / k)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l / t_m) * ((l / (t_m * (t_m * k))) / k));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(l / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

   6. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

   12. *-lowering-*.f64 47.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

5. Simplified 47.9%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

   2. times-frac N/A

      \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot t\right), \color{blue}{\left(k \cdot k\right)}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{k} \cdot k\right)\right)\right)\right) \]

   8. *-lowering-*.f64 57.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right) \]

7. Applied egg-rr 57.9%

   \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{k}}\right)\right) \]

   2. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\frac{\ell}{\left(t \cdot t\right) \cdot k}}{\color{blue}{k}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\left(\frac{\ell}{\left(t \cdot t\right) \cdot k}\right), \color{blue}{k}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(\left(t \cdot t\right) \cdot k\right)\right), k\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot k\right)\right)\right), k\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot k\right)\right)\right), k\right)\right) \]

   7. *-lowering-*.f64 66.3%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, k\right)\right)\right), k\right)\right) \]

9. Applied egg-rr 66.3%

   \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\frac{\ell}{t \cdot \left(t \cdot k\right)}}{k}} \]
10. Add Preprocessing

Alternative 14: 67.0% accurate, 32.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\ell}{t\_m} \cdot \frac{\ell}{\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot k\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (/ l t_m) (/ l (* (* t_m k) (* t_m k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / t_m) * (l / ((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 / t_m) * (l / ((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 / t_m) * (l / ((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 / t_m) * (l / ((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(Float64(l / t_m) * Float64(l / Float64(Float64(t_m * 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 / t_m) * (l / ((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[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

   6. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

   12. *-lowering-*.f64 47.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

5. Simplified 47.9%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

   2. times-frac N/A

      \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot t\right), \color{blue}{\left(k \cdot k\right)}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{k} \cdot k\right)\right)\right)\right) \]

   8. *-lowering-*.f64 57.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right) \]

7. Applied egg-rr 57.9%

   \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]
8. Step-by-step derivation

   1. unswap-sqr N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \left(\left(t \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\left(t \cdot k\right), \color{blue}{\left(t \cdot k\right)}\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \left(\color{blue}{t} \cdot k\right)\right)\right)\right) \]

   4. *-lowering-*.f64 64.7%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), \mathsf{*.f64}\left(t, \color{blue}{k}\right)\right)\right)\right) \]

9. Applied egg-rr 64.7%

   \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(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))))