Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.6% → 90.2%

Time: 16.4s

Alternatives: 19

Speedup: 12.5×

• 27.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.6% 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: 90.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{2 \cdot \ell}{t\_m \cdot k}\right)}^{-1} \cdot {\left(\frac{\ell}{k}\right)}^{-1}\right)}\\ \mathbf{elif}\;t\_m \leq 7.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + \frac{k \cdot k}{t\_m \cdot t\_m}\right)\right) \cdot \frac{\sin k \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left({t\_m}^{0.75} \cdot \frac{{t\_m}^{0.75}}{\ell}\right)}^{2}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.2e-55)
    (/
     1.0
     (*
      (* (sin k) (tan k))
      (* (pow (/ (* 2.0 l) (* t_m k)) -1.0) (pow (/ l k) -1.0))))
    (if (<= t_m 7.5e+80)
      (/
       2.0
       (/
        (*
         (* (tan k) (+ 2.0 (/ (* k k) (* t_m t_m))))
         (/ (* (sin k) (* t_m (* t_m t_m))) l))
        l))
      (/
       2.0
       (*
        (*
         (tan k)
         (* (sin k) (pow (* (pow t_m 0.75) (/ (pow t_m 0.75) l)) 2.0)))
        (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-55) {
		tmp = 1.0 / ((sin(k) * tan(k)) * (pow(((2.0 * l) / (t_m * k)), -1.0) * pow((l / k), -1.0)));
	} else if (t_m <= 7.5e+80) {
		tmp = 2.0 / (((tan(k) * (2.0 + ((k * k) / (t_m * t_m)))) * ((sin(k) * (t_m * (t_m * t_m))) / l)) / l);
	} else {
		tmp = 2.0 / ((tan(k) * (sin(k) * pow((pow(t_m, 0.75) * (pow(t_m, 0.75) / l)), 2.0))) * (1.0 + (1.0 + pow((k / t_m), 2.0))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4.2d-55) then
        tmp = 1.0d0 / ((sin(k) * tan(k)) * ((((2.0d0 * l) / (t_m * k)) ** (-1.0d0)) * ((l / k) ** (-1.0d0))))
    else if (t_m <= 7.5d+80) then
        tmp = 2.0d0 / (((tan(k) * (2.0d0 + ((k * k) / (t_m * t_m)))) * ((sin(k) * (t_m * (t_m * t_m))) / l)) / l)
    else
        tmp = 2.0d0 / ((tan(k) * (sin(k) * (((t_m ** 0.75d0) * ((t_m ** 0.75d0) / l)) ** 2.0d0))) * (1.0d0 + (1.0d0 + ((k / t_m) ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-55) {
		tmp = 1.0 / ((Math.sin(k) * Math.tan(k)) * (Math.pow(((2.0 * l) / (t_m * k)), -1.0) * Math.pow((l / k), -1.0)));
	} else if (t_m <= 7.5e+80) {
		tmp = 2.0 / (((Math.tan(k) * (2.0 + ((k * k) / (t_m * t_m)))) * ((Math.sin(k) * (t_m * (t_m * t_m))) / l)) / l);
	} else {
		tmp = 2.0 / ((Math.tan(k) * (Math.sin(k) * Math.pow((Math.pow(t_m, 0.75) * (Math.pow(t_m, 0.75) / l)), 2.0))) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4.2e-55:
		tmp = 1.0 / ((math.sin(k) * math.tan(k)) * (math.pow(((2.0 * l) / (t_m * k)), -1.0) * math.pow((l / k), -1.0)))
	elif t_m <= 7.5e+80:
		tmp = 2.0 / (((math.tan(k) * (2.0 + ((k * k) / (t_m * t_m)))) * ((math.sin(k) * (t_m * (t_m * t_m))) / l)) / l)
	else:
		tmp = 2.0 / ((math.tan(k) * (math.sin(k) * math.pow((math.pow(t_m, 0.75) * (math.pow(t_m, 0.75) / l)), 2.0))) * (1.0 + (1.0 + math.pow((k / t_m), 2.0))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.2e-55)
		tmp = Float64(1.0 / Float64(Float64(sin(k) * tan(k)) * Float64((Float64(Float64(2.0 * l) / Float64(t_m * k)) ^ -1.0) * (Float64(l / k) ^ -1.0))));
	elseif (t_m <= 7.5e+80)
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(2.0 + Float64(Float64(k * k) / Float64(t_m * t_m)))) * Float64(Float64(sin(k) * Float64(t_m * Float64(t_m * t_m))) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * (Float64((t_m ^ 0.75) * Float64((t_m ^ 0.75) / l)) ^ 2.0))) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4.2e-55)
		tmp = 1.0 / ((sin(k) * tan(k)) * ((((2.0 * l) / (t_m * k)) ^ -1.0) * ((l / k) ^ -1.0)));
	elseif (t_m <= 7.5e+80)
		tmp = 2.0 / (((tan(k) * (2.0 + ((k * k) / (t_m * t_m)))) * ((sin(k) * (t_m * (t_m * t_m))) / l)) / l);
	else
		tmp = 2.0 / ((tan(k) * (sin(k) * (((t_m ^ 0.75) * ((t_m ^ 0.75) / l)) ^ 2.0))) * (1.0 + (1.0 + ((k / t_m) ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.2e-55], N[(1.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(l / k), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.5e+80], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[(N[(k * k), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 0.75], $MachinePrecision] * N[(N[Power[t$95$m, 0.75], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.2000000000000003e-55

   1. Initial program 56.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 61.2

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

   5. Simplified 61.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. times-frac N/A

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

   7. Applied egg-rr 62.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. clear-num N/A

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

      6. inv-pow N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. unpow-prod-down N/A

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

      15. lower-*.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. lower-pow.f64 N/A

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

   9. Applied egg-rr 78.9%

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

   if 4.2000000000000003e-55 < t < 7.49999999999999994e80

   1. Initial program 65.9%

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

   4. Applied egg-rr 92.1%

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

   if 7.49999999999999994e80 < t

   1. Initial program 62.4%

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

      1. sqr-pow N/A

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

      2. times-frac N/A

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

      3. pow2 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval 85.7

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

   4. Applied egg-rr 85.7%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. sqr-pow N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\color{blue}{\frac{3}{4}}} \cdot \frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \color{blue}{\frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \frac{\color{blue}{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. metadata-eval 88.0

         \[\leadsto \frac{2}{\left(\left({\left({t}^{0.75} \cdot \frac{{t}^{\color{blue}{0.75}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied egg-rr 88.0%

      \[\leadsto \frac{2}{\left(\left({\color{blue}{\left({t}^{0.75} \cdot \frac{{t}^{0.75}}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{2 \cdot \ell}{t \cdot k}\right)}^{-1} \cdot {\left(\frac{\ell}{k}\right)}^{-1}\right)}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right) \cdot \frac{\sin k \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left({t}^{0.75} \cdot \frac{{t}^{0.75}}{\ell}\right)}^{2}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{2 \cdot \ell}{t\_m \cdot k}\right)}^{-1} \cdot {\left(\frac{\ell}{k}\right)}^{-1}\right)}\\ \mathbf{elif}\;t\_m \leq 7.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + \frac{k \cdot k}{t\_m \cdot t\_m}\right)\right) \cdot \frac{\sin k \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left({t\_m}^{0.75} \cdot {\left(\frac{{t\_m}^{1.125}}{\ell}\right)}^{2}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.2e-55)
    (/
     1.0
     (*
      (* (sin k) (tan k))
      (* (pow (/ (* 2.0 l) (* t_m k)) -1.0) (pow (/ l k) -1.0))))
    (if (<= t_m 7.5e+80)
      (/
       2.0
       (/
        (*
         (* (tan k) (+ 2.0 (/ (* k k) (* t_m t_m))))
         (/ (* (sin k) (* t_m (* t_m t_m))) l))
        l))
      (/
       2.0
       (*
        (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))
        (*
         (tan k)
         (* (sin k) (* (pow t_m 0.75) (pow (/ (pow t_m 1.125) l) 2.0))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-55) {
		tmp = 1.0 / ((sin(k) * tan(k)) * (pow(((2.0 * l) / (t_m * k)), -1.0) * pow((l / k), -1.0)));
	} else if (t_m <= 7.5e+80) {
		tmp = 2.0 / (((tan(k) * (2.0 + ((k * k) / (t_m * t_m)))) * ((sin(k) * (t_m * (t_m * t_m))) / l)) / l);
	} else {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t_m), 2.0))) * (tan(k) * (sin(k) * (pow(t_m, 0.75) * pow((pow(t_m, 1.125) / l), 2.0)))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4.2d-55) then
        tmp = 1.0d0 / ((sin(k) * tan(k)) * ((((2.0d0 * l) / (t_m * k)) ** (-1.0d0)) * ((l / k) ** (-1.0d0))))
    else if (t_m <= 7.5d+80) then
        tmp = 2.0d0 / (((tan(k) * (2.0d0 + ((k * k) / (t_m * t_m)))) * ((sin(k) * (t_m * (t_m * t_m))) / l)) / l)
    else
        tmp = 2.0d0 / ((1.0d0 + (1.0d0 + ((k / t_m) ** 2.0d0))) * (tan(k) * (sin(k) * ((t_m ** 0.75d0) * (((t_m ** 1.125d0) / l) ** 2.0d0)))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-55) {
		tmp = 1.0 / ((Math.sin(k) * Math.tan(k)) * (Math.pow(((2.0 * l) / (t_m * k)), -1.0) * Math.pow((l / k), -1.0)));
	} else if (t_m <= 7.5e+80) {
		tmp = 2.0 / (((Math.tan(k) * (2.0 + ((k * k) / (t_m * t_m)))) * ((Math.sin(k) * (t_m * (t_m * t_m))) / l)) / l);
	} else {
		tmp = 2.0 / ((1.0 + (1.0 + Math.pow((k / t_m), 2.0))) * (Math.tan(k) * (Math.sin(k) * (Math.pow(t_m, 0.75) * Math.pow((Math.pow(t_m, 1.125) / l), 2.0)))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4.2e-55:
		tmp = 1.0 / ((math.sin(k) * math.tan(k)) * (math.pow(((2.0 * l) / (t_m * k)), -1.0) * math.pow((l / k), -1.0)))
	elif t_m <= 7.5e+80:
		tmp = 2.0 / (((math.tan(k) * (2.0 + ((k * k) / (t_m * t_m)))) * ((math.sin(k) * (t_m * (t_m * t_m))) / l)) / l)
	else:
		tmp = 2.0 / ((1.0 + (1.0 + math.pow((k / t_m), 2.0))) * (math.tan(k) * (math.sin(k) * (math.pow(t_m, 0.75) * math.pow((math.pow(t_m, 1.125) / l), 2.0)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.2e-55)
		tmp = Float64(1.0 / Float64(Float64(sin(k) * tan(k)) * Float64((Float64(Float64(2.0 * l) / Float64(t_m * k)) ^ -1.0) * (Float64(l / k) ^ -1.0))));
	elseif (t_m <= 7.5e+80)
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(2.0 + Float64(Float64(k * k) / Float64(t_m * t_m)))) * Float64(Float64(sin(k) * Float64(t_m * Float64(t_m * t_m))) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0))) * Float64(tan(k) * Float64(sin(k) * Float64((t_m ^ 0.75) * (Float64((t_m ^ 1.125) / l) ^ 2.0))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4.2e-55)
		tmp = 1.0 / ((sin(k) * tan(k)) * ((((2.0 * l) / (t_m * k)) ^ -1.0) * ((l / k) ^ -1.0)));
	elseif (t_m <= 7.5e+80)
		tmp = 2.0 / (((tan(k) * (2.0 + ((k * k) / (t_m * t_m)))) * ((sin(k) * (t_m * (t_m * t_m))) / l)) / l);
	else
		tmp = 2.0 / ((1.0 + (1.0 + ((k / t_m) ^ 2.0))) * (tan(k) * (sin(k) * ((t_m ^ 0.75) * (((t_m ^ 1.125) / l) ^ 2.0)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.2e-55], N[(1.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(l / k), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.5e+80], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[(N[(k * k), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 0.75], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.125], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.2000000000000003e-55

   1. Initial program 56.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 61.2

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

   5. Simplified 61.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. times-frac N/A

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

   7. Applied egg-rr 62.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. clear-num N/A

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

      6. inv-pow N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. unpow-prod-down N/A

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

      15. lower-*.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. lower-pow.f64 N/A

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

   9. Applied egg-rr 78.9%

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

   if 4.2000000000000003e-55 < t < 7.49999999999999994e80

   1. Initial program 65.9%

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

   4. Applied egg-rr 92.1%

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

   if 7.49999999999999994e80 < t

   1. Initial program 62.4%

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

      1. sqr-pow N/A

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

      2. times-frac N/A

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

      3. pow2 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval 85.7

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

   4. Applied egg-rr 85.7%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. sqr-pow N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\color{blue}{\frac{3}{4}}} \cdot \frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \color{blue}{\frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \frac{\color{blue}{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. metadata-eval 88.0

         \[\leadsto \frac{2}{\left(\left({\left({t}^{0.75} \cdot \frac{{t}^{\color{blue}{0.75}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied egg-rr 88.0%

      \[\leadsto \frac{2}{\left(\left({\color{blue}{\left({t}^{0.75} \cdot \frac{{t}^{0.75}}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\color{blue}{{t}^{\frac{3}{4}}} \cdot \frac{{t}^{\frac{3}{4}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \frac{\color{blue}{{t}^{\frac{3}{4}}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \color{blue}{\frac{{t}^{\frac{3}{4}}}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left({t}^{\frac{3}{4}} \cdot \frac{{t}^{\frac{3}{4}}}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. unpow1 N/A

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left({\left({t}^{\frac{3}{4}} \cdot \frac{{t}^{\frac{3}{4}}}{\ell}\right)}^{1}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left({\left({\left({t}^{\frac{3}{4}} \cdot \frac{{t}^{\frac{3}{4}}}{\ell}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left({\left({\left({t}^{\frac{3}{4}} \cdot \frac{{t}^{\frac{3}{4}}}{\ell}\right)}^{\color{blue}{1}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. unpow1 N/A

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left({t}^{\frac{3}{4}} \cdot \frac{{t}^{\frac{3}{4}}}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left({t}^{\frac{3}{4}} \cdot \frac{{t}^{\frac{3}{4}}}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left({\color{blue}{\left(\frac{{t}^{\frac{3}{4}}}{\ell} \cdot {t}^{\frac{3}{4}}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{\frac{3}{4}}}{\ell} \cdot \color{blue}{{t}^{\frac{3}{4}}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. sqr-pow N/A

          \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{\frac{3}{4}}}{\ell} \cdot \color{blue}{\left({t}^{\left(\frac{\frac{3}{4}}{2}\right)} \cdot {t}^{\left(\frac{\frac{3}{4}}{2}\right)}\right)}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. associate-*r* N/A

          \[\leadsto \frac{2}{\left(\left({\color{blue}{\left(\left(\frac{{t}^{\frac{3}{4}}}{\ell} \cdot {t}^{\left(\frac{\frac{3}{4}}{2}\right)}\right) \cdot {t}^{\left(\frac{\frac{3}{4}}{2}\right)}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. unpow-prod-down N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\left({\left(\frac{{t}^{\frac{3}{4}}}{\ell} \cdot {t}^{\left(\frac{\frac{3}{4}}{2}\right)}\right)}^{2} \cdot {\left({t}^{\left(\frac{\frac{3}{4}}{2}\right)}\right)}^{2}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. pow2 N/A

          \[\leadsto \frac{2}{\left(\left(\left({\left(\frac{{t}^{\frac{3}{4}}}{\ell} \cdot {t}^{\left(\frac{\frac{3}{4}}{2}\right)}\right)}^{2} \cdot \color{blue}{\left({t}^{\left(\frac{\frac{3}{4}}{2}\right)} \cdot {t}^{\left(\frac{\frac{3}{4}}{2}\right)}\right)}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. sqr-pow N/A

          \[\leadsto \frac{2}{\left(\left(\left({\left(\frac{{t}^{\frac{3}{4}}}{\ell} \cdot {t}^{\left(\frac{\frac{3}{4}}{2}\right)}\right)}^{2} \cdot \color{blue}{{t}^{\frac{3}{4}}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      17. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left({\left(\frac{{t}^{\frac{3}{4}}}{\ell} \cdot {t}^{\left(\frac{\frac{3}{4}}{2}\right)}\right)}^{2} \cdot \color{blue}{{t}^{\frac{3}{4}}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\left({\left(\frac{{t}^{\frac{3}{4}}}{\ell} \cdot {t}^{\left(\frac{\frac{3}{4}}{2}\right)}\right)}^{2} \cdot {t}^{\frac{3}{4}}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Applied egg-rr 88.0%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left({\left(\frac{{t}^{0.75}}{\ell} \cdot {t}^{0.375}\right)}^{2} \cdot {t}^{0.75}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left({\left(\frac{\color{blue}{{t}^{\frac{3}{4}}}}{\ell} \cdot {t}^{\frac{3}{8}}\right)}^{2} \cdot {t}^{\frac{3}{4}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left({\left(\frac{{t}^{\frac{3}{4}}}{\ell} \cdot \color{blue}{{t}^{\frac{3}{8}}}\right)}^{2} \cdot {t}^{\frac{3}{4}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left({\left(\color{blue}{\frac{{t}^{\frac{3}{4}}}{\ell}} \cdot {t}^{\frac{3}{8}}\right)}^{2} \cdot {t}^{\frac{3}{4}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left({\color{blue}{\left(\frac{{t}^{\frac{3}{4}}}{\ell} \cdot {t}^{\frac{3}{8}}\right)}}^{2} \cdot {t}^{\frac{3}{4}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-pow.f64 88.0

         \[\leadsto \frac{2}{\left(\left(\left(\color{blue}{{\left(\frac{{t}^{0.75}}{\ell} \cdot {t}^{0.375}\right)}^{2}} \cdot {t}^{0.75}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left({\color{blue}{\left(\frac{{t}^{\frac{3}{4}}}{\ell} \cdot {t}^{\frac{3}{8}}\right)}}^{2} \cdot {t}^{\frac{3}{4}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left({\left(\color{blue}{\frac{{t}^{\frac{3}{4}}}{\ell}} \cdot {t}^{\frac{3}{8}}\right)}^{2} \cdot {t}^{\frac{3}{4}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\left(\left({\color{blue}{\left(\frac{{t}^{\frac{3}{4}} \cdot {t}^{\frac{3}{8}}}{\ell}\right)}}^{2} \cdot {t}^{\frac{3}{4}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left({\color{blue}{\left(\frac{{t}^{\frac{3}{4}} \cdot {t}^{\frac{3}{8}}}{\ell}\right)}}^{2} \cdot {t}^{\frac{3}{4}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left({\left(\frac{\color{blue}{{t}^{\frac{3}{4}}} \cdot {t}^{\frac{3}{8}}}{\ell}\right)}^{2} \cdot {t}^{\frac{3}{4}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left({\left(\frac{{t}^{\frac{3}{4}} \cdot \color{blue}{{t}^{\frac{3}{8}}}}{\ell}\right)}^{2} \cdot {t}^{\frac{3}{4}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. pow-prod-up N/A

          \[\leadsto \frac{2}{\left(\left(\left({\left(\frac{\color{blue}{{t}^{\left(\frac{3}{4} + \frac{3}{8}\right)}}}{\ell}\right)}^{2} \cdot {t}^{\frac{3}{4}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left({\left(\frac{\color{blue}{{t}^{\left(\frac{3}{4} + \frac{3}{8}\right)}}}{\ell}\right)}^{2} \cdot {t}^{\frac{3}{4}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. metadata-eval 88.0

          \[\leadsto \frac{2}{\left(\left(\left({\left(\frac{{t}^{\color{blue}{1.125}}}{\ell}\right)}^{2} \cdot {t}^{0.75}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   10. Applied egg-rr 88.0%

       \[\leadsto \frac{2}{\left(\left(\left(\color{blue}{{\left(\frac{{t}^{1.125}}{\ell}\right)}^{2}} \cdot {t}^{0.75}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{2 \cdot \ell}{t \cdot k}\right)}^{-1} \cdot {\left(\frac{\ell}{k}\right)}^{-1}\right)}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right) \cdot \frac{\sin k \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left({t}^{0.75} \cdot {\left(\frac{{t}^{1.125}}{\ell}\right)}^{2}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{2 \cdot \ell}{t\_m \cdot k}\right)}^{-1} \cdot {\left(\frac{\ell}{k}\right)}^{-1}\right)}\\ \mathbf{elif}\;t\_m \leq 5.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + \frac{k \cdot k}{t\_m \cdot t\_m}\right)\right) \cdot \frac{\sin k \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left({t\_m}^{1.5} \cdot \frac{1}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.2e-55)
    (/
     1.0
     (*
      (* (sin k) (tan k))
      (* (pow (/ (* 2.0 l) (* t_m k)) -1.0) (pow (/ l k) -1.0))))
    (if (<= t_m 5.5e+80)
      (/
       2.0
       (/
        (*
         (* (tan k) (+ 2.0 (/ (* k k) (* t_m t_m))))
         (/ (* (sin k) (* t_m (* t_m t_m))) l))
        l))
      (/
       2.0
       (*
        (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))
        (* (tan k) (* (sin k) (pow (* (pow t_m 1.5) (/ 1.0 l)) 2.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-55) {
		tmp = 1.0 / ((sin(k) * tan(k)) * (pow(((2.0 * l) / (t_m * k)), -1.0) * pow((l / k), -1.0)));
	} else if (t_m <= 5.5e+80) {
		tmp = 2.0 / (((tan(k) * (2.0 + ((k * k) / (t_m * t_m)))) * ((sin(k) * (t_m * (t_m * t_m))) / l)) / l);
	} else {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t_m), 2.0))) * (tan(k) * (sin(k) * pow((pow(t_m, 1.5) * (1.0 / l)), 2.0))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4.2d-55) then
        tmp = 1.0d0 / ((sin(k) * tan(k)) * ((((2.0d0 * l) / (t_m * k)) ** (-1.0d0)) * ((l / k) ** (-1.0d0))))
    else if (t_m <= 5.5d+80) then
        tmp = 2.0d0 / (((tan(k) * (2.0d0 + ((k * k) / (t_m * t_m)))) * ((sin(k) * (t_m * (t_m * t_m))) / l)) / l)
    else
        tmp = 2.0d0 / ((1.0d0 + (1.0d0 + ((k / t_m) ** 2.0d0))) * (tan(k) * (sin(k) * (((t_m ** 1.5d0) * (1.0d0 / l)) ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-55) {
		tmp = 1.0 / ((Math.sin(k) * Math.tan(k)) * (Math.pow(((2.0 * l) / (t_m * k)), -1.0) * Math.pow((l / k), -1.0)));
	} else if (t_m <= 5.5e+80) {
		tmp = 2.0 / (((Math.tan(k) * (2.0 + ((k * k) / (t_m * t_m)))) * ((Math.sin(k) * (t_m * (t_m * t_m))) / l)) / l);
	} else {
		tmp = 2.0 / ((1.0 + (1.0 + Math.pow((k / t_m), 2.0))) * (Math.tan(k) * (Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) * (1.0 / l)), 2.0))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4.2e-55:
		tmp = 1.0 / ((math.sin(k) * math.tan(k)) * (math.pow(((2.0 * l) / (t_m * k)), -1.0) * math.pow((l / k), -1.0)))
	elif t_m <= 5.5e+80:
		tmp = 2.0 / (((math.tan(k) * (2.0 + ((k * k) / (t_m * t_m)))) * ((math.sin(k) * (t_m * (t_m * t_m))) / l)) / l)
	else:
		tmp = 2.0 / ((1.0 + (1.0 + math.pow((k / t_m), 2.0))) * (math.tan(k) * (math.sin(k) * math.pow((math.pow(t_m, 1.5) * (1.0 / l)), 2.0))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.2e-55)
		tmp = Float64(1.0 / Float64(Float64(sin(k) * tan(k)) * Float64((Float64(Float64(2.0 * l) / Float64(t_m * k)) ^ -1.0) * (Float64(l / k) ^ -1.0))));
	elseif (t_m <= 5.5e+80)
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(2.0 + Float64(Float64(k * k) / Float64(t_m * t_m)))) * Float64(Float64(sin(k) * Float64(t_m * Float64(t_m * t_m))) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0))) * Float64(tan(k) * Float64(sin(k) * (Float64((t_m ^ 1.5) * Float64(1.0 / l)) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4.2e-55)
		tmp = 1.0 / ((sin(k) * tan(k)) * ((((2.0 * l) / (t_m * k)) ^ -1.0) * ((l / k) ^ -1.0)));
	elseif (t_m <= 5.5e+80)
		tmp = 2.0 / (((tan(k) * (2.0 + ((k * k) / (t_m * t_m)))) * ((sin(k) * (t_m * (t_m * t_m))) / l)) / l);
	else
		tmp = 2.0 / ((1.0 + (1.0 + ((k / t_m) ^ 2.0))) * (tan(k) * (sin(k) * (((t_m ^ 1.5) * (1.0 / l)) ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.2e-55], N[(1.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(l / k), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.5e+80], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[(N[(k * k), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.2000000000000003e-55

   1. Initial program 56.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 61.2

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

   5. Simplified 61.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. times-frac N/A

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

   7. Applied egg-rr 62.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. clear-num N/A

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

      6. inv-pow N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. unpow-prod-down N/A

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

      15. lower-*.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. lower-pow.f64 N/A

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

   9. Applied egg-rr 78.9%

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

   if 4.2000000000000003e-55 < t < 5.49999999999999967e80

   1. Initial program 65.9%

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

   4. Applied egg-rr 92.1%

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

   if 5.49999999999999967e80 < t

   1. Initial program 62.4%

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

      1. sqr-pow N/A

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

      2. times-frac N/A

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

      3. pow2 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval 85.7

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

   4. Applied egg-rr 85.7%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. div-inv N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 85.7

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

   6. Applied egg-rr 85.7%

      \[\leadsto \frac{2}{\left(\left({\color{blue}{\left({t}^{1.5} \cdot \frac{1}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{2 \cdot \ell}{t \cdot k}\right)}^{-1} \cdot {\left(\frac{\ell}{k}\right)}^{-1}\right)}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right) \cdot \frac{\sin k \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left({t}^{1.5} \cdot \frac{1}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{2 \cdot \ell}{t\_m \cdot k}\right)}^{-1} \cdot {\left(\frac{\ell}{k}\right)}^{-1}\right)}\\ \mathbf{elif}\;t\_m \leq 5.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + \frac{k \cdot k}{t\_m \cdot t\_m}\right)\right) \cdot \frac{\sin k \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.2e-55)
    (/
     1.0
     (*
      (* (sin k) (tan k))
      (* (pow (/ (* 2.0 l) (* t_m k)) -1.0) (pow (/ l k) -1.0))))
    (if (<= t_m 5.5e+80)
      (/
       2.0
       (/
        (*
         (* (tan k) (+ 2.0 (/ (* k k) (* t_m t_m))))
         (/ (* (sin k) (* t_m (* t_m t_m))) l))
        l))
      (/
       2.0
       (*
        (* (tan k) (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0)))
        (fma (/ k t_m) (/ k t_m) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-55) {
		tmp = 1.0 / ((sin(k) * tan(k)) * (pow(((2.0 * l) / (t_m * k)), -1.0) * pow((l / k), -1.0)));
	} else if (t_m <= 5.5e+80) {
		tmp = 2.0 / (((tan(k) * (2.0 + ((k * k) / (t_m * t_m)))) * ((sin(k) * (t_m * (t_m * t_m))) / l)) / l);
	} else {
		tmp = 2.0 / ((tan(k) * (sin(k) * pow((pow(t_m, 1.5) / l), 2.0))) * fma((k / t_m), (k / t_m), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.2e-55)
		tmp = Float64(1.0 / Float64(Float64(sin(k) * tan(k)) * Float64((Float64(Float64(2.0 * l) / Float64(t_m * k)) ^ -1.0) * (Float64(l / k) ^ -1.0))));
	elseif (t_m <= 5.5e+80)
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(2.0 + Float64(Float64(k * k) / Float64(t_m * t_m)))) * Float64(Float64(sin(k) * Float64(t_m * Float64(t_m * t_m))) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0))) * fma(Float64(k / t_m), Float64(k / t_m), 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.2e-55], N[(1.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(l / k), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.5e+80], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[(N[(k * k), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.2000000000000003e-55

   1. Initial program 56.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 61.2

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

   5. Simplified 61.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. times-frac N/A

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

   7. Applied egg-rr 62.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. clear-num N/A

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

      6. inv-pow N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. unpow-prod-down N/A

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

      15. lower-*.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. lower-pow.f64 N/A

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

   9. Applied egg-rr 78.9%

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

   if 4.2000000000000003e-55 < t < 5.49999999999999967e80

   1. Initial program 65.9%

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

   4. Applied egg-rr 92.1%

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

   if 5.49999999999999967e80 < t

   1. Initial program 62.4%

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

      1. sqr-pow N/A

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

      2. times-frac N/A

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

      3. pow2 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval 85.7

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

   4. Applied egg-rr 85.7%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. times-frac N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. lower-fma.f64 85.7

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

   6. Applied egg-rr 85.7%

      \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{2 \cdot \ell}{t \cdot k}\right)}^{-1} \cdot {\left(\frac{\ell}{k}\right)}^{-1}\right)}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right) \cdot \frac{\sin k \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{2 \cdot \ell}{t\_m \cdot k}\right)}^{-1} \cdot {\left(\frac{\ell}{k}\right)}^{-1}\right)}\\ \mathbf{elif}\;t\_m \leq 2.3 \cdot 10^{+80}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + \frac{k \cdot k}{t\_m \cdot t\_m}\right)\right) \cdot \frac{\sin k \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m \cdot \left(t\_m \cdot \sin k\right)}{\ell}\right)\right)}\\ \end{array} \end{array} \]

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 (<= t_m 4.2e-55)
    (/
     1.0
     (*
      (* (sin k) (tan k))
      (* (pow (/ (* 2.0 l) (* t_m k)) -1.0) (pow (/ l k) -1.0))))
    (if (<= t_m 2.3e+80)
      (/
       2.0
       (/
        (*
         (* (tan k) (+ 2.0 (/ (* k k) (* t_m t_m))))
         (/ (* (sin k) (* t_m (* t_m t_m))) l))
        l))
      (/
       2.0
       (*
        (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))
        (* (tan k) (* (/ t_m l) (/ (* t_m (* t_m (sin k))) l)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-55) {
		tmp = 1.0 / ((sin(k) * tan(k)) * (pow(((2.0 * l) / (t_m * k)), -1.0) * pow((l / k), -1.0)));
	} else if (t_m <= 2.3e+80) {
		tmp = 2.0 / (((tan(k) * (2.0 + ((k * k) / (t_m * t_m)))) * ((sin(k) * (t_m * (t_m * t_m))) / l)) / l);
	} else {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t_m), 2.0))) * (tan(k) * ((t_m / l) * ((t_m * (t_m * sin(k))) / l))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4.2d-55) then
        tmp = 1.0d0 / ((sin(k) * tan(k)) * ((((2.0d0 * l) / (t_m * k)) ** (-1.0d0)) * ((l / k) ** (-1.0d0))))
    else if (t_m <= 2.3d+80) then
        tmp = 2.0d0 / (((tan(k) * (2.0d0 + ((k * k) / (t_m * t_m)))) * ((sin(k) * (t_m * (t_m * t_m))) / l)) / l)
    else
        tmp = 2.0d0 / ((1.0d0 + (1.0d0 + ((k / t_m) ** 2.0d0))) * (tan(k) * ((t_m / l) * ((t_m * (t_m * sin(k))) / l))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-55) {
		tmp = 1.0 / ((Math.sin(k) * Math.tan(k)) * (Math.pow(((2.0 * l) / (t_m * k)), -1.0) * Math.pow((l / k), -1.0)));
	} else if (t_m <= 2.3e+80) {
		tmp = 2.0 / (((Math.tan(k) * (2.0 + ((k * k) / (t_m * t_m)))) * ((Math.sin(k) * (t_m * (t_m * t_m))) / l)) / l);
	} else {
		tmp = 2.0 / ((1.0 + (1.0 + Math.pow((k / t_m), 2.0))) * (Math.tan(k) * ((t_m / l) * ((t_m * (t_m * Math.sin(k))) / l))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4.2e-55:
		tmp = 1.0 / ((math.sin(k) * math.tan(k)) * (math.pow(((2.0 * l) / (t_m * k)), -1.0) * math.pow((l / k), -1.0)))
	elif t_m <= 2.3e+80:
		tmp = 2.0 / (((math.tan(k) * (2.0 + ((k * k) / (t_m * t_m)))) * ((math.sin(k) * (t_m * (t_m * t_m))) / l)) / l)
	else:
		tmp = 2.0 / ((1.0 + (1.0 + math.pow((k / t_m), 2.0))) * (math.tan(k) * ((t_m / l) * ((t_m * (t_m * math.sin(k))) / l))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.2e-55)
		tmp = Float64(1.0 / Float64(Float64(sin(k) * tan(k)) * Float64((Float64(Float64(2.0 * l) / Float64(t_m * k)) ^ -1.0) * (Float64(l / k) ^ -1.0))));
	elseif (t_m <= 2.3e+80)
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(2.0 + Float64(Float64(k * k) / Float64(t_m * t_m)))) * Float64(Float64(sin(k) * Float64(t_m * Float64(t_m * t_m))) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0))) * Float64(tan(k) * Float64(Float64(t_m / l) * Float64(Float64(t_m * Float64(t_m * sin(k))) / l)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4.2e-55)
		tmp = 1.0 / ((sin(k) * tan(k)) * ((((2.0 * l) / (t_m * k)) ^ -1.0) * ((l / k) ^ -1.0)));
	elseif (t_m <= 2.3e+80)
		tmp = 2.0 / (((tan(k) * (2.0 + ((k * k) / (t_m * t_m)))) * ((sin(k) * (t_m * (t_m * t_m))) / l)) / l);
	else
		tmp = 2.0 / ((1.0 + (1.0 + ((k / t_m) ^ 2.0))) * (tan(k) * ((t_m / l) * ((t_m * (t_m * sin(k))) / l))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.2e-55], N[(1.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(l / k), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.3e+80], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[(N[(k * k), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.2000000000000003e-55

   1. Initial program 56.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 61.2

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

   5. Simplified 61.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. times-frac N/A

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

   7. Applied egg-rr 62.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. clear-num N/A

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

      6. inv-pow N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. unpow-prod-down N/A

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

      15. lower-*.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. lower-pow.f64 N/A

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

   9. Applied egg-rr 78.9%

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

   if 4.2000000000000003e-55 < t < 2.30000000000000004e80

   1. Initial program 65.9%

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

   4. Applied egg-rr 92.1%

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

   if 2.30000000000000004e80 < t

   1. Initial program 62.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 78.5

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

   4. Applied egg-rr 78.5%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{2 \cdot \ell}{t \cdot k}\right)}^{-1} \cdot {\left(\frac{\ell}{k}\right)}^{-1}\right)}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+80}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right) \cdot \frac{\sin k \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{t\_m \cdot k}{2 \cdot \ell} \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;t\_m \leq 2.3 \cdot 10^{+80}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + \frac{k \cdot k}{t\_m \cdot t\_m}\right)\right) \cdot \frac{\sin k \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m \cdot \left(t\_m \cdot \sin k\right)}{\ell}\right)\right)}\\ \end{array} \end{array} \]

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 (<= t_m 4.2e-55)
    (/ 1.0 (* (* (sin k) (tan k)) (* (/ (* t_m k) (* 2.0 l)) (/ k l))))
    (if (<= t_m 2.3e+80)
      (/
       2.0
       (/
        (*
         (* (tan k) (+ 2.0 (/ (* k k) (* t_m t_m))))
         (/ (* (sin k) (* t_m (* t_m t_m))) l))
        l))
      (/
       2.0
       (*
        (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))
        (* (tan k) (* (/ t_m l) (/ (* t_m (* t_m (sin k))) l)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-55) {
		tmp = 1.0 / ((sin(k) * tan(k)) * (((t_m * k) / (2.0 * l)) * (k / l)));
	} else if (t_m <= 2.3e+80) {
		tmp = 2.0 / (((tan(k) * (2.0 + ((k * k) / (t_m * t_m)))) * ((sin(k) * (t_m * (t_m * t_m))) / l)) / l);
	} else {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t_m), 2.0))) * (tan(k) * ((t_m / l) * ((t_m * (t_m * sin(k))) / l))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4.2d-55) then
        tmp = 1.0d0 / ((sin(k) * tan(k)) * (((t_m * k) / (2.0d0 * l)) * (k / l)))
    else if (t_m <= 2.3d+80) then
        tmp = 2.0d0 / (((tan(k) * (2.0d0 + ((k * k) / (t_m * t_m)))) * ((sin(k) * (t_m * (t_m * t_m))) / l)) / l)
    else
        tmp = 2.0d0 / ((1.0d0 + (1.0d0 + ((k / t_m) ** 2.0d0))) * (tan(k) * ((t_m / l) * ((t_m * (t_m * sin(k))) / l))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-55) {
		tmp = 1.0 / ((Math.sin(k) * Math.tan(k)) * (((t_m * k) / (2.0 * l)) * (k / l)));
	} else if (t_m <= 2.3e+80) {
		tmp = 2.0 / (((Math.tan(k) * (2.0 + ((k * k) / (t_m * t_m)))) * ((Math.sin(k) * (t_m * (t_m * t_m))) / l)) / l);
	} else {
		tmp = 2.0 / ((1.0 + (1.0 + Math.pow((k / t_m), 2.0))) * (Math.tan(k) * ((t_m / l) * ((t_m * (t_m * Math.sin(k))) / l))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4.2e-55:
		tmp = 1.0 / ((math.sin(k) * math.tan(k)) * (((t_m * k) / (2.0 * l)) * (k / l)))
	elif t_m <= 2.3e+80:
		tmp = 2.0 / (((math.tan(k) * (2.0 + ((k * k) / (t_m * t_m)))) * ((math.sin(k) * (t_m * (t_m * t_m))) / l)) / l)
	else:
		tmp = 2.0 / ((1.0 + (1.0 + math.pow((k / t_m), 2.0))) * (math.tan(k) * ((t_m / l) * ((t_m * (t_m * math.sin(k))) / l))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.2e-55)
		tmp = Float64(1.0 / Float64(Float64(sin(k) * tan(k)) * Float64(Float64(Float64(t_m * k) / Float64(2.0 * l)) * Float64(k / l))));
	elseif (t_m <= 2.3e+80)
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(2.0 + Float64(Float64(k * k) / Float64(t_m * t_m)))) * Float64(Float64(sin(k) * Float64(t_m * Float64(t_m * t_m))) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0))) * Float64(tan(k) * Float64(Float64(t_m / l) * Float64(Float64(t_m * Float64(t_m * sin(k))) / l)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4.2e-55)
		tmp = 1.0 / ((sin(k) * tan(k)) * (((t_m * k) / (2.0 * l)) * (k / l)));
	elseif (t_m <= 2.3e+80)
		tmp = 2.0 / (((tan(k) * (2.0 + ((k * k) / (t_m * t_m)))) * ((sin(k) * (t_m * (t_m * t_m))) / l)) / l);
	else
		tmp = 2.0 / ((1.0 + (1.0 + ((k / t_m) ^ 2.0))) * (tan(k) * ((t_m / l) * ((t_m * (t_m * sin(k))) / l))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.2e-55], N[(1.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$m * k), $MachinePrecision] / N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.3e+80], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[(N[(k * k), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.2000000000000003e-55

   1. Initial program 56.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 61.2

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

   5. Simplified 61.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. times-frac N/A

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

   7. Applied egg-rr 62.4%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 78.9

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

   9. Applied egg-rr 78.9%

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

   if 4.2000000000000003e-55 < t < 2.30000000000000004e80

   1. Initial program 65.9%

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

   4. Applied egg-rr 92.1%

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

   if 2.30000000000000004e80 < t

   1. Initial program 62.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 78.5

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

   4. Applied egg-rr 78.5%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{t \cdot k}{2 \cdot \ell} \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+80}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right) \cdot \frac{\sin k \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 350000000:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot \left(t\_m \cdot k\right)}}{k}\\ \mathbf{elif}\;k \leq 2.65 \cdot 10^{+141}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \frac{\cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left(k \cdot \frac{t\_m \cdot k}{2 \cdot \left(\ell \cdot \ell\right)}\right)}\\ \end{array} \end{array} \]

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 350000000.0)
    (* l (/ (/ (/ l t_m) (* t_m (* t_m k))) k))
    (if (<= k 2.65e+141)
      (*
       (* 2.0 l)
       (* l (/ (cos k) (* (- 0.5 (* 0.5 (cos (+ k k)))) (* t_m (* k k))))))
      (/ 1.0 (* (* (sin k) (tan k)) (* k (/ (* t_m k) (* 2.0 (* 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 <= 350000000.0) {
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	} else if (k <= 2.65e+141) {
		tmp = (2.0 * l) * (l * (cos(k) / ((0.5 - (0.5 * cos((k + k)))) * (t_m * (k * k)))));
	} else {
		tmp = 1.0 / ((sin(k) * tan(k)) * (k * ((t_m * k) / (2.0 * (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 <= 350000000.0d0) then
        tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k)
    else if (k <= 2.65d+141) then
        tmp = (2.0d0 * l) * (l * (cos(k) / ((0.5d0 - (0.5d0 * cos((k + k)))) * (t_m * (k * k)))))
    else
        tmp = 1.0d0 / ((sin(k) * tan(k)) * (k * ((t_m * k) / (2.0d0 * (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 <= 350000000.0) {
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	} else if (k <= 2.65e+141) {
		tmp = (2.0 * l) * (l * (Math.cos(k) / ((0.5 - (0.5 * Math.cos((k + k)))) * (t_m * (k * k)))));
	} else {
		tmp = 1.0 / ((Math.sin(k) * Math.tan(k)) * (k * ((t_m * k) / (2.0 * (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 <= 350000000.0:
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k)
	elif k <= 2.65e+141:
		tmp = (2.0 * l) * (l * (math.cos(k) / ((0.5 - (0.5 * math.cos((k + k)))) * (t_m * (k * k)))))
	else:
		tmp = 1.0 / ((math.sin(k) * math.tan(k)) * (k * ((t_m * k) / (2.0 * (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 <= 350000000.0)
		tmp = Float64(l * Float64(Float64(Float64(l / t_m) / Float64(t_m * Float64(t_m * k))) / k));
	elseif (k <= 2.65e+141)
		tmp = Float64(Float64(2.0 * l) * Float64(l * Float64(cos(k) / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))) * Float64(t_m * Float64(k * k))))));
	else
		tmp = Float64(1.0 / Float64(Float64(sin(k) * tan(k)) * Float64(k * Float64(Float64(t_m * k) / Float64(2.0 * 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 <= 350000000.0)
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	elseif (k <= 2.65e+141)
		tmp = (2.0 * l) * (l * (cos(k) / ((0.5 - (0.5 * cos((k + k)))) * (t_m * (k * k)))));
	else
		tmp = 1.0 / ((sin(k) * tan(k)) * (k * ((t_m * k) / (2.0 * (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, 350000000.0], N[(l * N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.65e+141], N[(N[(2.0 * l), $MachinePrecision] * N[(l * N[(N[Cos[k], $MachinePrecision] / N[(N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(k * N[(N[(t$95$m * k), $MachinePrecision] / N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3.5e8

   1. Initial program 58.8%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 50.5

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

   5. Simplified 50.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 56.7

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 62.0

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

   7. Applied egg-rr 62.0%

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 67.9

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

   9. Applied egg-rr 67.9%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/r* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. associate-/r* N/A

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

       9. lower-/.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-*.f64 73.3

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 73.3

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

   11. Applied egg-rr 73.3%

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

   if 3.5e8 < k < 2.65e141

   1. Initial program 46.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 78.5

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

   5. Simplified 78.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 86.8

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

   7. Applied egg-rr 86.7%

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

   if 2.65e141 < k

   1. Initial program 64.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 70.1

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

   5. Simplified 70.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. times-frac N/A

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

   7. Applied egg-rr 70.1%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 83.6

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 83.6

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 83.6

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

   9. Applied egg-rr 83.6%

      \[\leadsto \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \color{blue}{\left(k \cdot \frac{k \cdot t}{2 \cdot \left(\ell \cdot \ell\right)}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 350000000:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot k\right)}}{k}\\ \mathbf{elif}\;k \leq 2.65 \cdot 10^{+141}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \frac{\cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left(k \cdot \frac{t \cdot k}{2 \cdot \left(\ell \cdot \ell\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.1 \cdot 10^{+47}:\\ \;\;\;\;\frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{t\_m \cdot k}{2 \cdot \ell} \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot \left(t\_m \cdot k\right)}}{k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.1e+47)
    (/ 1.0 (* (* (sin k) (tan k)) (* (/ (* t_m k) (* 2.0 l)) (/ k l))))
    (* l (/ (/ (/ l t_m) (* t_m (* t_m k))) k)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.1e+47) {
		tmp = 1.0 / ((sin(k) * tan(k)) * (((t_m * k) / (2.0 * l)) * (k / l)));
	} else {
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.1d+47) then
        tmp = 1.0d0 / ((sin(k) * tan(k)) * (((t_m * k) / (2.0d0 * l)) * (k / l)))
    else
        tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.1e+47) {
		tmp = 1.0 / ((Math.sin(k) * Math.tan(k)) * (((t_m * k) / (2.0 * l)) * (k / l)));
	} else {
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.1e+47:
		tmp = 1.0 / ((math.sin(k) * math.tan(k)) * (((t_m * k) / (2.0 * l)) * (k / l)))
	else:
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.1e+47)
		tmp = Float64(1.0 / Float64(Float64(sin(k) * tan(k)) * Float64(Float64(Float64(t_m * k) / Float64(2.0 * l)) * Float64(k / l))));
	else
		tmp = Float64(l * Float64(Float64(Float64(l / t_m) / Float64(t_m * Float64(t_m * k))) / k));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.1e+47)
		tmp = 1.0 / ((sin(k) * tan(k)) * (((t_m * k) / (2.0 * l)) * (k / l)));
	else
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.1e+47], N[(1.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$m * k), $MachinePrecision] / N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.1e47

   1. Initial program 57.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 61.4

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

   5. Simplified 61.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. times-frac N/A

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

   7. Applied egg-rr 62.4%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 78.9

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

   9. Applied egg-rr 78.9%

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

   if 2.1e47 < t

   1. Initial program 61.9%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 52.6

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

   5. Simplified 52.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 55.8

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 62.5

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

   7. Applied egg-rr 62.5%

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 75.3

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

   9. Applied egg-rr 75.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/r* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. associate-/r* N/A

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

       9. lower-/.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-*.f64 81.6

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 81.6

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

   11. Applied egg-rr 81.6%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{+47}:\\ \;\;\;\;\frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{t \cdot k}{2 \cdot \ell} \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot k\right)}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 350000000:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot \left(t\_m \cdot k\right)}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(t\_m \cdot k\right)}}{\sin k \cdot \tan k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 350000000.0)
    (* l (/ (/ (/ l t_m) (* t_m (* t_m k))) k))
    (/ (/ (* 2.0 (* l l)) (* k (* t_m k))) (* (sin k) (tan k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 350000000.0) {
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	} else {
		tmp = ((2.0 * (l * l)) / (k * (t_m * k))) / (sin(k) * tan(k));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 350000000.0d0) then
        tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k)
    else
        tmp = ((2.0d0 * (l * l)) / (k * (t_m * k))) / (sin(k) * tan(k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 350000000.0) {
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	} else {
		tmp = ((2.0 * (l * l)) / (k * (t_m * k))) / (Math.sin(k) * Math.tan(k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 350000000.0:
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k)
	else:
		tmp = ((2.0 * (l * l)) / (k * (t_m * k))) / (math.sin(k) * math.tan(k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 350000000.0)
		tmp = Float64(l * Float64(Float64(Float64(l / t_m) / Float64(t_m * Float64(t_m * k))) / k));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(l * l)) / Float64(k * Float64(t_m * k))) / Float64(sin(k) * tan(k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 350000000.0)
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	else
		tmp = ((2.0 * (l * l)) / (k * (t_m * k))) / (sin(k) * tan(k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 350000000.0], N[(l * N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.5e8

   1. Initial program 58.8%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 50.5

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

   5. Simplified 50.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 56.7

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 62.0

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

   7. Applied egg-rr 62.0%

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 67.9

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

   9. Applied egg-rr 67.9%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/r* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. associate-/r* N/A

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

       9. lower-/.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-*.f64 73.3

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 73.3

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

   11. Applied egg-rr 73.3%

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

   if 3.5e8 < k

   1. Initial program 57.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 73.3

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

   5. Simplified 73.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      10. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{\frac{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\color{blue}{\cos k \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right)}}} \]

      15. times-frac N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t \cdot \left(k \cdot k\right)}{2 \cdot \left(\ell \cdot \ell\right)}}} \]

   7. Applied egg-rr 73.2%

      \[\leadsto \color{blue}{\frac{1}{\left(\sin k \cdot \tan k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \left(\ell \cdot 2\right)}}} \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{1}{\left(\color{blue}{\sin k} \cdot \tan k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \left(\ell \cdot 2\right)}} \]

      2. lift-tan.f64 N/A

         \[\leadsto \frac{1}{\left(\sin k \cdot \color{blue}{\tan k}\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \left(\ell \cdot 2\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \left(\ell \cdot 2\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \frac{t \cdot \color{blue}{\left(k \cdot k\right)}}{\ell \cdot \left(\ell \cdot 2\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \frac{\color{blue}{t \cdot \left(k \cdot k\right)}}{\ell \cdot \left(\ell \cdot 2\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \color{blue}{\left(\ell \cdot 2\right)}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\color{blue}{\ell \cdot \left(\ell \cdot 2\right)}}} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \left(\ell \cdot 2\right)}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \left(\ell \cdot 2\right)} \cdot \left(\sin k \cdot \tan k\right)}} \]

      10. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \left(\ell \cdot 2\right)}}}{\sin k \cdot \tan k}} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1}{\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \left(\ell \cdot 2\right)}}}}{\sin k \cdot \tan k} \]

      12. clear-num N/A

          \[\leadsto \frac{\color{blue}{\frac{\ell \cdot \left(\ell \cdot 2\right)}{t \cdot \left(k \cdot k\right)}}}{\sin k \cdot \tan k} \]

      13. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \left(\ell \cdot 2\right)}{t \cdot \left(k \cdot k\right)}}{\sin k \cdot \tan k}} \]

   9. Applied egg-rr 73.3%

      \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)}}{\sin k \cdot \tan k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 350000000:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot k\right)}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(t \cdot k\right)}}{\sin k \cdot \tan k}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 350000000:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot \left(t\_m \cdot k\right)}}{k}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left(k \cdot \left(t\_m \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 350000000.0)
    (* l (/ (/ (/ l t_m) (* t_m (* t_m k))) k))
    (* (* 2.0 (* l l)) (/ 1.0 (* (* (sin k) (tan k)) (* k (* t_m k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 350000000.0) {
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	} else {
		tmp = (2.0 * (l * l)) * (1.0 / ((sin(k) * tan(k)) * (k * (t_m * k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 350000000.0d0) then
        tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k)
    else
        tmp = (2.0d0 * (l * l)) * (1.0d0 / ((sin(k) * tan(k)) * (k * (t_m * k))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 350000000.0) {
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	} else {
		tmp = (2.0 * (l * l)) * (1.0 / ((Math.sin(k) * Math.tan(k)) * (k * (t_m * k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 350000000.0:
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k)
	else:
		tmp = (2.0 * (l * l)) * (1.0 / ((math.sin(k) * math.tan(k)) * (k * (t_m * k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 350000000.0)
		tmp = Float64(l * Float64(Float64(Float64(l / t_m) / Float64(t_m * Float64(t_m * k))) / k));
	else
		tmp = Float64(Float64(2.0 * Float64(l * l)) * Float64(1.0 / Float64(Float64(sin(k) * tan(k)) * Float64(k * Float64(t_m * k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 350000000.0)
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	else
		tmp = (2.0 * (l * l)) * (1.0 / ((sin(k) * tan(k)) * (k * (t_m * k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 350000000.0], N[(l * N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.5e8

   1. Initial program 58.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 50.5

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 50.5%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 56.7

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      14. associate-*l* N/A

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)} \cdot \ell \]

      16. lower-*.f64 62.0

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

   7. Applied egg-rr 62.0%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

      3. lower-*.f64 67.9

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot k\right)} \cdot k\right)\right)} \cdot \ell \]

   9. Applied egg-rr 67.9%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot k\right)} \cdot k\right)\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right)}} \cdot \ell \]

       4. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{t \cdot \left(\left(t \cdot k\right) \cdot k\right)}} \cdot \ell \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t}}{\color{blue}{t \cdot \left(\left(t \cdot k\right) \cdot k\right)}} \cdot \ell \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t}}{t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \cdot \ell \]

       7. associate-*r* N/A

          \[\leadsto \frac{\frac{\ell}{t}}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right) \cdot k}} \cdot \ell \]

       8. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot k\right)}}{k}} \cdot \ell \]

       9. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot k\right)}}{k}} \cdot \ell \]

       10. lower-/.f64 N/A

           \[\leadsto \frac{\color{blue}{\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot k\right)}}}{k} \cdot \ell \]

       11. lower-/.f64 N/A

           \[\leadsto \frac{\frac{\color{blue}{\frac{\ell}{t}}}{t \cdot \left(t \cdot k\right)}}{k} \cdot \ell \]

       12. lower-*.f64 73.3

           \[\leadsto \frac{\frac{\frac{\ell}{t}}{\color{blue}{t \cdot \left(t \cdot k\right)}}}{k} \cdot \ell \]

       13. lift-*.f64 N/A

           \[\leadsto \frac{\frac{\frac{\ell}{t}}{t \cdot \color{blue}{\left(t \cdot k\right)}}}{k} \cdot \ell \]

       14. *-commutative N/A

           \[\leadsto \frac{\frac{\frac{\ell}{t}}{t \cdot \color{blue}{\left(k \cdot t\right)}}}{k} \cdot \ell \]

       15. lower-*.f64 73.3

           \[\leadsto \frac{\frac{\frac{\ell}{t}}{t \cdot \color{blue}{\left(k \cdot t\right)}}}{k} \cdot \ell \]

   11. Applied egg-rr 73.3%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot t\right)}}{k}} \cdot \ell \]

   if 3.5e8 < k

   1. Initial program 57.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left({k}^{2} \cdot t\right)} \]

      13. lower-sin.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\color{blue}{\sin k}}^{2} \cdot \left({k}^{2} \cdot t\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      16. unpow2 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      17. lower-*.f64 73.3

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Simplified 73.3%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right)} \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\cos k}}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\color{blue}{\sin k}}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      10. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{\frac{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\color{blue}{\cos k \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right)}}} \]

      15. times-frac N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t \cdot \left(k \cdot k\right)}{2 \cdot \left(\ell \cdot \ell\right)}}} \]

   7. Applied egg-rr 73.2%

      \[\leadsto \color{blue}{\frac{1}{\left(\sin k \cdot \tan k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \left(\ell \cdot 2\right)}}} \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{1}{\left(\color{blue}{\sin k} \cdot \tan k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \left(\ell \cdot 2\right)}} \]

      2. lift-tan.f64 N/A

         \[\leadsto \frac{1}{\left(\sin k \cdot \color{blue}{\tan k}\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \left(\ell \cdot 2\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \left(\ell \cdot 2\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \frac{t \cdot \color{blue}{\left(k \cdot k\right)}}{\ell \cdot \left(\ell \cdot 2\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \frac{\color{blue}{t \cdot \left(k \cdot k\right)}}{\ell \cdot \left(\ell \cdot 2\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \color{blue}{\left(\ell \cdot 2\right)}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\color{blue}{\ell \cdot \left(\ell \cdot 2\right)}}} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \left(\ell \cdot 2\right)}}} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \left(\ell \cdot 2\right)}}} \]

      10. associate-*r/ N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\ell \cdot \left(\ell \cdot 2\right)}}} \]

      11. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \left(\ell \cdot \left(\ell \cdot 2\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \left(\ell \cdot \left(\ell \cdot 2\right)\right)} \]

   9. Applied egg-rr 73.2%

      \[\leadsto \color{blue}{\frac{1}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 350000000:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot k\right)}}{k}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left(k \cdot \left(t \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 230000:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot \left(t\_m \cdot k\right)}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right)}{\left(k \cdot k\right) \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 230000.0)
    (* l (/ (/ (/ l t_m) (* t_m (* t_m k))) k))
    (/ (* (cos k) (* 2.0 (* l l))) (* (* k k) (* t_m (* k k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 230000.0) {
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	} else {
		tmp = (cos(k) * (2.0 * (l * l))) / ((k * k) * (t_m * (k * k)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 230000.0d0) then
        tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k)
    else
        tmp = (cos(k) * (2.0d0 * (l * l))) / ((k * k) * (t_m * (k * k)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 230000.0) {
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	} else {
		tmp = (Math.cos(k) * (2.0 * (l * l))) / ((k * k) * (t_m * (k * k)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 230000.0:
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k)
	else:
		tmp = (math.cos(k) * (2.0 * (l * l))) / ((k * k) * (t_m * (k * k)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 230000.0)
		tmp = Float64(l * Float64(Float64(Float64(l / t_m) / Float64(t_m * Float64(t_m * k))) / k));
	else
		tmp = Float64(Float64(cos(k) * Float64(2.0 * Float64(l * l))) / Float64(Float64(k * k) * Float64(t_m * Float64(k * k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 230000.0)
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	else
		tmp = (cos(k) * (2.0 * (l * l))) / ((k * k) * (t_m * (k * k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 230000.0], N[(l * N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[k], $MachinePrecision] * N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.3e5

   1. Initial program 58.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 50.3

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 50.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 56.5

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      14. associate-*l* N/A

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)} \cdot \ell \]

      16. lower-*.f64 61.8

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

   7. Applied egg-rr 61.8%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

      3. lower-*.f64 67.7

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot k\right)} \cdot k\right)\right)} \cdot \ell \]

   9. Applied egg-rr 67.7%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot k\right)} \cdot k\right)\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right)}} \cdot \ell \]

       4. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{t \cdot \left(\left(t \cdot k\right) \cdot k\right)}} \cdot \ell \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t}}{\color{blue}{t \cdot \left(\left(t \cdot k\right) \cdot k\right)}} \cdot \ell \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t}}{t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \cdot \ell \]

       7. associate-*r* N/A

          \[\leadsto \frac{\frac{\ell}{t}}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right) \cdot k}} \cdot \ell \]

       8. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot k\right)}}{k}} \cdot \ell \]

       9. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot k\right)}}{k}} \cdot \ell \]

       10. lower-/.f64 N/A

           \[\leadsto \frac{\color{blue}{\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot k\right)}}}{k} \cdot \ell \]

       11. lower-/.f64 N/A

           \[\leadsto \frac{\frac{\color{blue}{\frac{\ell}{t}}}{t \cdot \left(t \cdot k\right)}}{k} \cdot \ell \]

       12. lower-*.f64 73.2

           \[\leadsto \frac{\frac{\frac{\ell}{t}}{\color{blue}{t \cdot \left(t \cdot k\right)}}}{k} \cdot \ell \]

       13. lift-*.f64 N/A

           \[\leadsto \frac{\frac{\frac{\ell}{t}}{t \cdot \color{blue}{\left(t \cdot k\right)}}}{k} \cdot \ell \]

       14. *-commutative N/A

           \[\leadsto \frac{\frac{\frac{\ell}{t}}{t \cdot \color{blue}{\left(k \cdot t\right)}}}{k} \cdot \ell \]

       15. lower-*.f64 73.2

           \[\leadsto \frac{\frac{\frac{\ell}{t}}{t \cdot \color{blue}{\left(k \cdot t\right)}}}{k} \cdot \ell \]

   11. Applied egg-rr 73.2%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot t\right)}}{k}} \cdot \ell \]

   if 2.3e5 < k

   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. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left({k}^{2} \cdot t\right)} \]

      13. lower-sin.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\color{blue}{\sin k}}^{2} \cdot \left({k}^{2} \cdot t\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      16. unpow2 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      17. lower-*.f64 72.3

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Simplified 72.3%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      2. lower-*.f64 64.1

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

   8. Simplified 64.1%

      \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 230000:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot k\right)}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right)}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.1% accurate, 7.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{+37}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot \left(t\_m \cdot k\right)}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(k \cdot k\right) \cdot \frac{t\_m \cdot \left(k \cdot k\right)}{\ell \cdot \left(2 \cdot \ell\right)}}\\ \end{array} \end{array} \]

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 8.5e+37)
    (* l (/ (/ (/ l t_m) (* t_m (* t_m k))) k))
    (/ 1.0 (* (* k k) (/ (* t_m (* k k)) (* l (* 2.0 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 <= 8.5e+37) {
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	} else {
		tmp = 1.0 / ((k * k) * ((t_m * (k * k)) / (l * (2.0 * 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 <= 8.5d+37) then
        tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k)
    else
        tmp = 1.0d0 / ((k * k) * ((t_m * (k * k)) / (l * (2.0d0 * 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 <= 8.5e+37) {
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	} else {
		tmp = 1.0 / ((k * k) * ((t_m * (k * k)) / (l * (2.0 * 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 <= 8.5e+37:
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k)
	else:
		tmp = 1.0 / ((k * k) * ((t_m * (k * k)) / (l * (2.0 * 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 <= 8.5e+37)
		tmp = Float64(l * Float64(Float64(Float64(l / t_m) / Float64(t_m * Float64(t_m * k))) / k));
	else
		tmp = Float64(1.0 / Float64(Float64(k * k) * Float64(Float64(t_m * Float64(k * k)) / Float64(l * Float64(2.0 * 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 <= 8.5e+37)
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	else
		tmp = 1.0 / ((k * k) * ((t_m * (k * k)) / (l * (2.0 * 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, 8.5e+37], N[(l * N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(l * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.4999999999999999e37

   1. Initial program 58.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 50.5

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 50.5%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 56.6

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      14. associate-*l* N/A

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)} \cdot \ell \]

      16. lower-*.f64 61.8

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

   7. Applied egg-rr 61.8%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

      3. lower-*.f64 67.6

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot k\right)} \cdot k\right)\right)} \cdot \ell \]

   9. Applied egg-rr 67.6%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot k\right)} \cdot k\right)\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right)}} \cdot \ell \]

       4. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{t \cdot \left(\left(t \cdot k\right) \cdot k\right)}} \cdot \ell \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t}}{\color{blue}{t \cdot \left(\left(t \cdot k\right) \cdot k\right)}} \cdot \ell \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t}}{t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \cdot \ell \]

       7. associate-*r* N/A

          \[\leadsto \frac{\frac{\ell}{t}}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right) \cdot k}} \cdot \ell \]

       8. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot k\right)}}{k}} \cdot \ell \]

       9. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot k\right)}}{k}} \cdot \ell \]

       10. lower-/.f64 N/A

           \[\leadsto \frac{\color{blue}{\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot k\right)}}}{k} \cdot \ell \]

       11. lower-/.f64 N/A

           \[\leadsto \frac{\frac{\color{blue}{\frac{\ell}{t}}}{t \cdot \left(t \cdot k\right)}}{k} \cdot \ell \]

       12. lower-*.f64 72.9

           \[\leadsto \frac{\frac{\frac{\ell}{t}}{\color{blue}{t \cdot \left(t \cdot k\right)}}}{k} \cdot \ell \]

       13. lift-*.f64 N/A

           \[\leadsto \frac{\frac{\frac{\ell}{t}}{t \cdot \color{blue}{\left(t \cdot k\right)}}}{k} \cdot \ell \]

       14. *-commutative N/A

           \[\leadsto \frac{\frac{\frac{\ell}{t}}{t \cdot \color{blue}{\left(k \cdot t\right)}}}{k} \cdot \ell \]

       15. lower-*.f64 72.9

           \[\leadsto \frac{\frac{\frac{\ell}{t}}{t \cdot \color{blue}{\left(k \cdot t\right)}}}{k} \cdot \ell \]

   11. Applied egg-rr 72.9%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot t\right)}}{k}} \cdot \ell \]

   if 8.4999999999999999e37 < k

   1. Initial program 57.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left({k}^{2} \cdot t\right)} \]

      13. lower-sin.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\color{blue}{\sin k}}^{2} \cdot \left({k}^{2} \cdot t\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      16. unpow2 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      17. lower-*.f64 73.1

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Simplified 73.1%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right)} \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\cos k}}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\color{blue}{\sin k}}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      10. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{\frac{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\color{blue}{\cos k \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right)}}} \]

      15. times-frac N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t \cdot \left(k \cdot k\right)}{2 \cdot \left(\ell \cdot \ell\right)}}} \]

   7. Applied egg-rr 73.0%

      \[\leadsto \color{blue}{\frac{1}{\left(\sin k \cdot \tan k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \left(\ell \cdot 2\right)}}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{1}{\color{blue}{{k}^{2}} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \left(\ell \cdot 2\right)}} \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \left(\ell \cdot 2\right)}} \]

      2. lower-*.f64 62.4

         \[\leadsto \frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \left(\ell \cdot 2\right)}} \]

   10. Simplified 62.4%

       \[\leadsto \frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \left(\ell \cdot 2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{+37}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot k\right)}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \left(2 \cdot \ell\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.1% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \left(k \cdot \left(t\_m \cdot k\right)\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-236}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot t\_m\right)\right)}\\ \mathbf{elif}\;k \leq 2.65 \cdot 10^{-25}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{t\_m}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t\_2}}{t\_m}\\ \end{array} \end{array} \end{array} \]

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 (* k (* t_m k)))))
   (*
    t_s
    (if (<= k 1.2e-236)
      (* (/ l k) (/ l (* t_m (* k (* t_m t_m)))))
      (if (<= k 2.65e-25) (* l (/ (/ l t_m) t_2)) (/ (/ (* l l) t_2) 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 t_2 = t_m * (k * (t_m * k));
	double tmp;
	if (k <= 1.2e-236) {
		tmp = (l / k) * (l / (t_m * (k * (t_m * t_m))));
	} else if (k <= 2.65e-25) {
		tmp = l * ((l / t_m) / t_2);
	} else {
		tmp = ((l * l) / t_2) / 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) :: t_2
    real(8) :: tmp
    t_2 = t_m * (k * (t_m * k))
    if (k <= 1.2d-236) then
        tmp = (l / k) * (l / (t_m * (k * (t_m * t_m))))
    else if (k <= 2.65d-25) then
        tmp = l * ((l / t_m) / t_2)
    else
        tmp = ((l * l) / t_2) / 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 t_2 = t_m * (k * (t_m * k));
	double tmp;
	if (k <= 1.2e-236) {
		tmp = (l / k) * (l / (t_m * (k * (t_m * t_m))));
	} else if (k <= 2.65e-25) {
		tmp = l * ((l / t_m) / t_2);
	} else {
		tmp = ((l * l) / t_2) / 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):
	t_2 = t_m * (k * (t_m * k))
	tmp = 0
	if k <= 1.2e-236:
		tmp = (l / k) * (l / (t_m * (k * (t_m * t_m))))
	elif k <= 2.65e-25:
		tmp = l * ((l / t_m) / t_2)
	else:
		tmp = ((l * l) / t_2) / 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)
	t_2 = Float64(t_m * Float64(k * Float64(t_m * k)))
	tmp = 0.0
	if (k <= 1.2e-236)
		tmp = Float64(Float64(l / k) * Float64(l / Float64(t_m * Float64(k * Float64(t_m * t_m)))));
	elseif (k <= 2.65e-25)
		tmp = Float64(l * Float64(Float64(l / t_m) / t_2));
	else
		tmp = Float64(Float64(Float64(l * l) / t_2) / 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)
	t_2 = t_m * (k * (t_m * k));
	tmp = 0.0;
	if (k <= 1.2e-236)
		tmp = (l / k) * (l / (t_m * (k * (t_m * t_m))));
	elseif (k <= 2.65e-25)
		tmp = l * ((l / t_m) / t_2);
	else
		tmp = ((l * l) / t_2) / 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_] := Block[{t$95$2 = N[(t$95$m * N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1.2e-236], N[(N[(l / k), $MachinePrecision] * N[(l / N[(t$95$m * N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.65e-25], N[(l * N[(N[(l / t$95$m), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / t$95$2), $MachinePrecision] / t$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 1.2000000000000001e-236

   1. Initial program 57.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 49.7

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 49.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot k}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot k} \cdot \frac{\ell}{k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot k} \cdot \frac{\ell}{k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot k}} \cdot \frac{\ell}{k} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot k} \cdot \frac{\ell}{k} \]

      10. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot k\right)}} \cdot \frac{\ell}{k} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot k\right)}} \cdot \frac{\ell}{k} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}} \cdot \frac{\ell}{k} \]

      13. lower-/.f64 70.1

          \[\leadsto \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot k\right)} \cdot \color{blue}{\frac{\ell}{k}} \]

   7. Applied egg-rr 70.1%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot k\right)} \cdot \frac{\ell}{k}} \]

   if 1.2000000000000001e-236 < k < 2.6499999999999998e-25

   1. Initial program 65.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 54.2

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 54.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 60.0

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      14. associate-*l* N/A

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)} \cdot \ell \]

      16. lower-*.f64 65.5

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

   7. Applied egg-rr 65.5%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

      3. lower-*.f64 79.5

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot k\right)} \cdot k\right)\right)} \cdot \ell \]

   9. Applied egg-rr 79.5%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot k\right)} \cdot k\right)\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right)}} \cdot \ell \]

       4. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{t \cdot \left(\left(t \cdot k\right) \cdot k\right)}} \cdot \ell \]

       5. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{t \cdot \left(\left(t \cdot k\right) \cdot k\right)}} \cdot \ell \]

       6. lower-/.f64 84.7

          \[\leadsto \frac{\color{blue}{\frac{\ell}{t}}}{t \cdot \left(\left(t \cdot k\right) \cdot k\right)} \cdot \ell \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t}}{t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \cdot \ell \]

       8. *-commutative N/A

          \[\leadsto \frac{\frac{\ell}{t}}{t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}} \cdot \ell \]

       9. lower-*.f64 84.7

          \[\leadsto \frac{\frac{\ell}{t}}{t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}} \cdot \ell \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{\frac{\ell}{t}}{t \cdot \left(k \cdot \color{blue}{\left(t \cdot k\right)}\right)} \cdot \ell \]

       11. *-commutative N/A

           \[\leadsto \frac{\frac{\ell}{t}}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \cdot \ell \]

       12. lower-*.f64 84.7

           \[\leadsto \frac{\frac{\ell}{t}}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \cdot \ell \]

   11. Applied egg-rr 84.7%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot \left(k \cdot t\right)\right)}} \cdot \ell \]

   if 2.6499999999999998e-25 < k

   1. Initial program 55.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 50.9

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 50.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 51.6

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      14. associate-*l* N/A

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)} \cdot \ell \]

      16. lower-*.f64 55.2

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

   7. Applied egg-rr 55.2%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

      3. lower-*.f64 53.7

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot k\right)} \cdot k\right)\right)} \cdot \ell \]

   9. Applied egg-rr 53.7%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot k\right)} \cdot k\right)\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right)}} \cdot \ell \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right)}} \cdot \ell \]

       5. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right)}} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right)}} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{t \cdot \left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right)} \]

       8. *-commutative N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right) \cdot t}} \]

       9. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t \cdot \left(\left(t \cdot k\right) \cdot k\right)}}{t}} \]

       10. lower-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t \cdot \left(\left(t \cdot k\right) \cdot k\right)}}{t}} \]

       11. lower-/.f64 62.1

           \[\leadsto \frac{\color{blue}{\frac{\ell \cdot \ell}{t \cdot \left(\left(t \cdot k\right) \cdot k\right)}}}{t} \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{\frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}}}{t} \]

       13. *-commutative N/A

           \[\leadsto \frac{\frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}}}{t} \]

       14. lower-*.f64 62.1

           \[\leadsto \frac{\frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}}}{t} \]

       15. lift-*.f64 N/A

           \[\leadsto \frac{\frac{\ell \cdot \ell}{t \cdot \left(k \cdot \color{blue}{\left(t \cdot k\right)}\right)}}{t} \]

       16. *-commutative N/A

           \[\leadsto \frac{\frac{\ell \cdot \ell}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)}}{t} \]

       17. lower-*.f64 62.1

           \[\leadsto \frac{\frac{\ell \cdot \ell}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)}}{t} \]

   11. Applied egg-rr 62.1%

       \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t \cdot \left(k \cdot \left(k \cdot t\right)\right)}}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-236}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot t\right)\right)}\\ \mathbf{elif}\;k \leq 2.65 \cdot 10^{-25}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{t}}{t \cdot \left(k \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t \cdot \left(k \cdot \left(t \cdot k\right)\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 67.5% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.65 \cdot 10^{-25}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot \left(t\_m \cdot k\right)}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot k\right)\right)}}{t\_m}\\ \end{array} \end{array} \]

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 2.65e-25)
    (* l (/ (/ (/ l t_m) (* t_m (* t_m k))) k))
    (/ (/ (* l l) (* t_m (* k (* t_m 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 <= 2.65e-25) {
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	} else {
		tmp = ((l * l) / (t_m * (k * (t_m * 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 <= 2.65d-25) then
        tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k)
    else
        tmp = ((l * l) / (t_m * (k * (t_m * 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 <= 2.65e-25) {
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	} else {
		tmp = ((l * l) / (t_m * (k * (t_m * 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 <= 2.65e-25:
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k)
	else:
		tmp = ((l * l) / (t_m * (k * (t_m * 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 <= 2.65e-25)
		tmp = Float64(l * Float64(Float64(Float64(l / t_m) / Float64(t_m * Float64(t_m * k))) / k));
	else
		tmp = Float64(Float64(Float64(l * l) / Float64(t_m * Float64(k * Float64(t_m * 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 <= 2.65e-25)
		tmp = l * (((l / t_m) / (t_m * (t_m * k))) / k);
	else
		tmp = ((l * l) / (t_m * (k * (t_m * 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, 2.65e-25], N[(l * N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / N[(t$95$m * N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.6499999999999998e-25

   1. Initial program 59.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 51.0

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 51.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 57.3

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      14. associate-*l* N/A

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)} \cdot \ell \]

      16. lower-*.f64 62.8

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

   7. Applied egg-rr 62.8%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

      3. lower-*.f64 68.8

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot k\right)} \cdot k\right)\right)} \cdot \ell \]

   9. Applied egg-rr 68.8%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot k\right)} \cdot k\right)\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right)}} \cdot \ell \]

       4. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{t \cdot \left(\left(t \cdot k\right) \cdot k\right)}} \cdot \ell \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t}}{\color{blue}{t \cdot \left(\left(t \cdot k\right) \cdot k\right)}} \cdot \ell \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t}}{t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \cdot \ell \]

       7. associate-*r* N/A

          \[\leadsto \frac{\frac{\ell}{t}}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right) \cdot k}} \cdot \ell \]

       8. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot k\right)}}{k}} \cdot \ell \]

       9. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot k\right)}}{k}} \cdot \ell \]

       10. lower-/.f64 N/A

           \[\leadsto \frac{\color{blue}{\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot k\right)}}}{k} \cdot \ell \]

       11. lower-/.f64 N/A

           \[\leadsto \frac{\frac{\color{blue}{\frac{\ell}{t}}}{t \cdot \left(t \cdot k\right)}}{k} \cdot \ell \]

       12. lower-*.f64 74.0

           \[\leadsto \frac{\frac{\frac{\ell}{t}}{\color{blue}{t \cdot \left(t \cdot k\right)}}}{k} \cdot \ell \]

       13. lift-*.f64 N/A

           \[\leadsto \frac{\frac{\frac{\ell}{t}}{t \cdot \color{blue}{\left(t \cdot k\right)}}}{k} \cdot \ell \]

       14. *-commutative N/A

           \[\leadsto \frac{\frac{\frac{\ell}{t}}{t \cdot \color{blue}{\left(k \cdot t\right)}}}{k} \cdot \ell \]

       15. lower-*.f64 74.0

           \[\leadsto \frac{\frac{\frac{\ell}{t}}{t \cdot \color{blue}{\left(k \cdot t\right)}}}{k} \cdot \ell \]

   11. Applied egg-rr 74.0%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot t\right)}}{k}} \cdot \ell \]

   if 2.6499999999999998e-25 < k

   1. Initial program 55.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 50.9

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 50.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 51.6

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      14. associate-*l* N/A

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)} \cdot \ell \]

      16. lower-*.f64 55.2

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

   7. Applied egg-rr 55.2%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

      3. lower-*.f64 53.7

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot k\right)} \cdot k\right)\right)} \cdot \ell \]

   9. Applied egg-rr 53.7%

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot k\right)} \cdot k\right)\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right)}} \cdot \ell \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right)}} \cdot \ell \]

       5. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right)}} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right)}} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{t \cdot \left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right)} \]

       8. *-commutative N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right) \cdot t}} \]

       9. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t \cdot \left(\left(t \cdot k\right) \cdot k\right)}}{t}} \]

       10. lower-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t \cdot \left(\left(t \cdot k\right) \cdot k\right)}}{t}} \]

       11. lower-/.f64 62.1

           \[\leadsto \frac{\color{blue}{\frac{\ell \cdot \ell}{t \cdot \left(\left(t \cdot k\right) \cdot k\right)}}}{t} \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{\frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}}}{t} \]

       13. *-commutative N/A

           \[\leadsto \frac{\frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}}}{t} \]

       14. lower-*.f64 62.1

           \[\leadsto \frac{\frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}}}{t} \]

       15. lift-*.f64 N/A

           \[\leadsto \frac{\frac{\ell \cdot \ell}{t \cdot \left(k \cdot \color{blue}{\left(t \cdot k\right)}\right)}}{t} \]

       16. *-commutative N/A

           \[\leadsto \frac{\frac{\ell \cdot \ell}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)}}{t} \]

       17. lower-*.f64 62.1

           \[\leadsto \frac{\frac{\ell \cdot \ell}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)}}{t} \]

   11. Applied egg-rr 62.1%

       \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t \cdot \left(k \cdot \left(k \cdot t\right)\right)}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.65 \cdot 10^{-25}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot \left(t \cdot k\right)}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t \cdot \left(k \cdot \left(t \cdot k\right)\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 66.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 8.2 \cdot 10^{-139}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot t\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 8.2e-139)
    (* (/ l k) (/ l (* t_m (* k (* t_m t_m)))))
    (* (/ l t_m) (/ l (* t_m (* t_m (* k k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 8.2e-139) {
		tmp = (l / k) * (l / (t_m * (k * (t_m * t_m))));
	} else {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 8.2d-139) then
        tmp = (l / k) * (l / (t_m * (k * (t_m * t_m))))
    else
        tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 8.2e-139) {
		tmp = (l / k) * (l / (t_m * (k * (t_m * t_m))));
	} else {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 8.2e-139:
		tmp = (l / k) * (l / (t_m * (k * (t_m * t_m))))
	else:
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 8.2e-139)
		tmp = Float64(Float64(l / k) * Float64(l / Float64(t_m * Float64(k * Float64(t_m * t_m)))));
	else
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(t_m * Float64(t_m * Float64(k * k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 8.2e-139)
		tmp = (l / k) * (l / (t_m * (k * (t_m * t_m))));
	else
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 8.2e-139], N[(N[(l / k), $MachinePrecision] * N[(l / N[(t$95$m * N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.20000000000000028e-139

   1. Initial program 60.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 50.1

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 50.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot k}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot k} \cdot \frac{\ell}{k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot k} \cdot \frac{\ell}{k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot k}} \cdot \frac{\ell}{k} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot k} \cdot \frac{\ell}{k} \]

      10. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot k\right)}} \cdot \frac{\ell}{k} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot k\right)}} \cdot \frac{\ell}{k} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}} \cdot \frac{\ell}{k} \]

      13. lower-/.f64 73.2

          \[\leadsto \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot k\right)} \cdot \color{blue}{\frac{\ell}{k}} \]

   7. Applied egg-rr 73.2%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot k\right)} \cdot \frac{\ell}{k}} \]

   if 8.20000000000000028e-139 < k

   1. Initial program 55.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 52.6

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{t}} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \]

      11. lower-*.f64 65.0

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

   7. Applied egg-rr 65.0%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.2 \cdot 10^{-139}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 68.1% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-162}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 5e-162)
    (* l (/ l (* t_m (* k (* t_m (* t_m k))))))
    (* (/ l t_m) (/ l (* t_m (* t_m (* k k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5e-162) {
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))));
	} else {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5d-162) then
        tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))))
    else
        tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5e-162) {
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))));
	} else {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 5e-162:
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))))
	else:
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 5e-162)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(k * Float64(t_m * Float64(t_m * k))))));
	else
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(t_m * Float64(t_m * Float64(k * k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 5e-162)
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))));
	else
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 5e-162], N[(l * N[(l / N[(t$95$m * N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.00000000000000014e-162

   1. Initial program 60.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 49.8

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 49.8%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 56.0

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      14. associate-*l* N/A

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)} \cdot \ell \]

      16. lower-*.f64 62.0

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

   7. Applied egg-rr 62.0%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot \left(t \cdot k\right)\right) \cdot k\right)}} \cdot \ell \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot \left(t \cdot k\right)\right) \cdot k\right)}} \cdot \ell \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{t \cdot \left(\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot k\right)} \cdot \ell \]

      5. lower-*.f64 72.3

         \[\leadsto \frac{\ell}{t \cdot \left(\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot k\right)} \cdot \ell \]

   9. Applied egg-rr 72.3%

      \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot \left(t \cdot k\right)\right) \cdot k\right)}} \cdot \ell \]

   if 5.00000000000000014e-162 < k

   1. Initial program 55.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 53.0

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 53.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{t}} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \]

      11. lower-*.f64 64.8

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

   7. Applied egg-rr 64.8%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-162}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 66.1% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* l (/ l (* t_m (* k (* t_m (* t_m k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (t_m * (k * (t_m * (t_m * k))))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (l * (l / (t_m * (k * (t_m * (t_m * k))))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (t_m * (k * (t_m * (t_m * k))))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * (l / (t_m * (k * (t_m * (t_m * k))))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(l / Float64(t_m * Float64(k * Float64(t_m * Float64(t_m * k)))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (l * (l / (t_m * (k * (t_m * (t_m * k))))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(l * N[(l / N[(t$95$m * N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   6. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   12. lower-*.f64 51.0

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Simplified 51.0%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   5. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   8. lower-/.f64 55.9

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

   11. associate-*l* N/A

       \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

   14. associate-*l* N/A

       \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)} \cdot \ell \]

   16. lower-*.f64 60.9

       \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

7. Applied egg-rr 60.9%

   \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \ell} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

   2. associate-*r* N/A

      \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot \left(t \cdot k\right)\right) \cdot k\right)}} \cdot \ell \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot \left(t \cdot k\right)\right) \cdot k\right)}} \cdot \ell \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{t \cdot \left(\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot k\right)} \cdot \ell \]

   5. lower-*.f64 66.8

      \[\leadsto \frac{\ell}{t \cdot \left(\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot k\right)} \cdot \ell \]

9. Applied egg-rr 66.8%

   \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot \left(t \cdot k\right)\right) \cdot k\right)}} \cdot \ell \]

10. Final simplification 66.8%

    \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)} \]
11. Add Preprocessing

Alternative 18: 65.1% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot \left(t\_m \cdot k\right)\right)\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* l (/ l (* t_m (* t_m (* k (* t_m k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (t_m * (t_m * (k * (t_m * k))))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (l * (l / (t_m * (t_m * (k * (t_m * k))))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (t_m * (t_m * (k * (t_m * k))))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * (l / (t_m * (t_m * (k * (t_m * k))))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(l / Float64(t_m * Float64(t_m * Float64(k * Float64(t_m * k)))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (l * (l / (t_m * (t_m * (k * (t_m * k))))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(l * N[(l / N[(t$95$m * N[(t$95$m * N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   6. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   12. lower-*.f64 51.0

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Simplified 51.0%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   5. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   8. lower-/.f64 55.9

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

   11. associate-*l* N/A

       \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

   14. associate-*l* N/A

       \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)} \cdot \ell \]

   16. lower-*.f64 60.9

       \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

7. Applied egg-rr 60.9%

   \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \ell} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

   3. lower-*.f64 65.0

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot k\right)} \cdot k\right)\right)} \cdot \ell \]

9. Applied egg-rr 65.0%

   \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right)} \cdot \ell \]

10. Final simplification 65.0%

    \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)} \]
11. Add Preprocessing

Alternative 19: 62.1% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* l (/ l (* t_m (* t_m (* t_m (* k k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (t_m * (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 * (l / (t_m * (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 * (l / (t_m * (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 * (l / (t_m * (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(l * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k)))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (l * (l / (t_m * (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[(l * N[(l / N[(t$95$m * N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   6. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   12. lower-*.f64 51.0

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Simplified 51.0%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   5. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   8. lower-/.f64 55.9

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

   11. associate-*l* N/A

       \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

   14. associate-*l* N/A

       \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)} \cdot \ell \]

   16. lower-*.f64 60.9

       \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

7. Applied egg-rr 60.9%

   \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot \ell} \]

8. Final simplification 60.9%

   \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(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))))