Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.0% → 94.9%

Time: 13.4s

Alternatives: 23

Speedup: 9.6×

• 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: 55.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.25 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \sin k\right) \cdot \left(k \cdot \left(\frac{k}{\ell} \cdot t\_m\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m} \cdot \ell}{\left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right) \cdot \sin k}}{\left(\tan k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.25e-18)
    (/ 2.0 (/ (* (* (tan k) (sin k)) (* k (* (/ k l) t_m))) l))
    (/
     (/ (* (/ 2.0 t_m) l) (* (+ (pow (/ k t_m) 2.0) 2.0) (sin k)))
     (* (* (tan k) t_m) (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.25e-18) {
		tmp = 2.0 / (((tan(k) * sin(k)) * (k * ((k / l) * t_m))) / l);
	} else {
		tmp = (((2.0 / t_m) * l) / ((pow((k / t_m), 2.0) + 2.0) * sin(k))) / ((tan(k) * t_m) * (t_m / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.25d-18) then
        tmp = 2.0d0 / (((tan(k) * sin(k)) * (k * ((k / l) * t_m))) / l)
    else
        tmp = (((2.0d0 / t_m) * l) / ((((k / t_m) ** 2.0d0) + 2.0d0) * sin(k))) / ((tan(k) * t_m) * (t_m / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.25e-18) {
		tmp = 2.0 / (((Math.tan(k) * Math.sin(k)) * (k * ((k / l) * t_m))) / l);
	} else {
		tmp = (((2.0 / t_m) * l) / ((Math.pow((k / t_m), 2.0) + 2.0) * Math.sin(k))) / ((Math.tan(k) * t_m) * (t_m / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.25e-18:
		tmp = 2.0 / (((math.tan(k) * math.sin(k)) * (k * ((k / l) * t_m))) / l)
	else:
		tmp = (((2.0 / t_m) * l) / ((math.pow((k / t_m), 2.0) + 2.0) * math.sin(k))) / ((math.tan(k) * t_m) * (t_m / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.25e-18)
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * sin(k)) * Float64(k * Float64(Float64(k / l) * t_m))) / l));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 / t_m) * l) / Float64(Float64((Float64(k / t_m) ^ 2.0) + 2.0) * sin(k))) / Float64(Float64(tan(k) * t_m) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.25e-18)
		tmp = 2.0 / (((tan(k) * sin(k)) * (k * ((k / l) * t_m))) / l);
	else
		tmp = (((2.0 / t_m) * l) / ((((k / t_m) ^ 2.0) + 2.0) * sin(k))) / ((tan(k) * t_m) * (t_m / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.25e-18], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(k * N[(N[(k / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.25000000000000009e-18

   1. Initial program 56.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 79.4

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

   5. Applied rewrites 79.4%

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

   7. Applied rewrites 78.2%

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

   9. Applied rewrites 83.1%

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

   if 1.25000000000000009e-18 < t

   1. Initial program 73.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 75.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. associate-+l+ N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

   6. Applied rewrites 91.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 96.8%

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

Alternative 2: 94.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \sin k\right) \cdot \left(k \cdot \left(\frac{k}{\ell} \cdot t\_m\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\left(\left(\tan k \cdot \frac{t\_m}{\ell}\right) \cdot t\_m\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right)\right)\right)}\\ \end{array} \end{array} \]

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 5.8e-12)
    (/ 2.0 (/ (* (* (tan k) (sin k)) (* k (* (/ k l) t_m))) l))
    (/
     2.0
     (*
      (/ t_m l)
      (*
       (* (* (tan k) (/ t_m l)) t_m)
       (* (sin k) (+ (pow (/ k t_m) 2.0) 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 <= 5.8e-12) {
		tmp = 2.0 / (((tan(k) * sin(k)) * (k * ((k / l) * t_m))) / l);
	} else {
		tmp = 2.0 / ((t_m / l) * (((tan(k) * (t_m / l)) * t_m) * (sin(k) * (pow((k / t_m), 2.0) + 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 <= 5.8d-12) then
        tmp = 2.0d0 / (((tan(k) * sin(k)) * (k * ((k / l) * t_m))) / l)
    else
        tmp = 2.0d0 / ((t_m / l) * (((tan(k) * (t_m / l)) * t_m) * (sin(k) * (((k / t_m) ** 2.0d0) + 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 <= 5.8e-12) {
		tmp = 2.0 / (((Math.tan(k) * Math.sin(k)) * (k * ((k / l) * t_m))) / l);
	} else {
		tmp = 2.0 / ((t_m / l) * (((Math.tan(k) * (t_m / l)) * t_m) * (Math.sin(k) * (Math.pow((k / t_m), 2.0) + 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 <= 5.8e-12:
		tmp = 2.0 / (((math.tan(k) * math.sin(k)) * (k * ((k / l) * t_m))) / l)
	else:
		tmp = 2.0 / ((t_m / l) * (((math.tan(k) * (t_m / l)) * t_m) * (math.sin(k) * (math.pow((k / t_m), 2.0) + 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 <= 5.8e-12)
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * sin(k)) * Float64(k * Float64(Float64(k / l) * t_m))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(Float64(tan(k) * Float64(t_m / l)) * t_m) * Float64(sin(k) * Float64((Float64(k / t_m) ^ 2.0) + 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 <= 5.8e-12)
		tmp = 2.0 / (((tan(k) * sin(k)) * (k * ((k / l) * t_m))) / l);
	else
		tmp = 2.0 / ((t_m / l) * (((tan(k) * (t_m / l)) * t_m) * (sin(k) * (((k / t_m) ^ 2.0) + 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, 5.8e-12], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(k * N[(N[(k / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.8000000000000003e-12

   1. Initial program 57.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 79.5

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

   5. Applied rewrites 79.5%

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

   7. Applied rewrites 78.3%

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

   9. Applied rewrites 83.1%

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

   if 5.8000000000000003e-12 < t

   1. Initial program 74.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 75.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. associate-+l+ N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

   6. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 95.0

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

   8. Applied rewrites 95.0%

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

Alternative 3: 77.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m} \cdot \ell}{2 \cdot k}}{\left(\tan k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{elif}\;k \leq 0.0275:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right)}{\ell} \cdot k, k, \frac{t\_m \cdot t\_m}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell} \cdot t\_m\right) \cdot \frac{\tan k \cdot \sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(\sin k \cdot \tan k\right) \cdot \frac{t\_m}{\ell}}{\ell} \cdot k\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.5e-83)
    (/ (/ (* (/ 2.0 t_m) l) (* 2.0 k)) (* (* (tan k) t_m) (/ t_m l)))
    (if (<= k 0.0275)
      (/
       2.0
       (*
        (/ t_m l)
        (*
         (fma
          (* (/ (fma 0.3333333333333333 (* t_m t_m) 1.0) l) k)
          k
          (* (/ (* t_m t_m) l) 2.0))
         (* k k))))
      (if (<= k 1.4e+154)
        (/ 2.0 (* (* (/ (* k k) l) t_m) (/ (* (tan k) (sin k)) l)))
        (/ 2.0 (* (* (/ (* (* (sin k) (tan k)) (/ t_m l)) l) 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 <= 1.5e-83) {
		tmp = (((2.0 / t_m) * l) / (2.0 * k)) / ((tan(k) * t_m) * (t_m / l));
	} else if (k <= 0.0275) {
		tmp = 2.0 / ((t_m / l) * (fma(((fma(0.3333333333333333, (t_m * t_m), 1.0) / l) * k), k, (((t_m * t_m) / l) * 2.0)) * (k * k)));
	} else if (k <= 1.4e+154) {
		tmp = 2.0 / ((((k * k) / l) * t_m) * ((tan(k) * sin(k)) / l));
	} else {
		tmp = 2.0 / (((((sin(k) * tan(k)) * (t_m / l)) / l) * 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 <= 1.5e-83)
		tmp = Float64(Float64(Float64(Float64(2.0 / t_m) * l) / Float64(2.0 * k)) / Float64(Float64(tan(k) * t_m) * Float64(t_m / l)));
	elseif (k <= 0.0275)
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(fma(Float64(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) / l) * k), k, Float64(Float64(Float64(t_m * t_m) / l) * 2.0)) * Float64(k * k))));
	elseif (k <= 1.4e+154)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) / l) * t_m) * Float64(Float64(tan(k) * sin(k)) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(sin(k) * tan(k)) * Float64(t_m / l)) / l) * k) * k));
	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[k, 1.5e-83], N[(N[(N[(N[(2.0 / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(2.0 * k), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.0275], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.4e+154], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if k < 1.50000000000000005e-83

   1. Initial program 62.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 65.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. associate-+l+ N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

   6. Applied rewrites 79.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 85.6%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 81.3

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

   11. Applied rewrites 81.3%

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

   if 1.50000000000000005e-83 < k < 0.0275000000000000001

   1. Initial program 59.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 78.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. associate-+l+ N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

   6. Applied rewrites 82.2%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 99.6%

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

   if 0.0275000000000000001 < k < 1.4e154

   1. Initial program 57.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 81.5

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

   5. Applied rewrites 81.5%

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

   7. Applied rewrites 81.7%

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

   if 1.4e154 < k

   1. Initial program 58.2%

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

   3. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 84.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{\frac{{\sin k}^{2}}{\ell}}{\ell \cdot \cos k} \cdot \mathsf{fma}\left(2, \frac{\frac{{t}^{3}}{k}}{k}, t\right)\right) \cdot k\right) \cdot k}} \]

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 82.0%

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

   10. Applied rewrites 84.6%

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

Alternative 4: 77.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m} \cdot \ell}{2 \cdot k}}{\left(\tan k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{elif}\;k \leq 0.0275:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right)}{\ell} \cdot k, k, \frac{t\_m \cdot t\_m}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell} \cdot t\_m\right) \cdot \left(\frac{\sin k}{\ell} \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(\sin k \cdot \tan k\right) \cdot \frac{t\_m}{\ell}}{\ell} \cdot k\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.5e-83)
    (/ (/ (* (/ 2.0 t_m) l) (* 2.0 k)) (* (* (tan k) t_m) (/ t_m l)))
    (if (<= k 0.0275)
      (/
       2.0
       (*
        (/ t_m l)
        (*
         (fma
          (* (/ (fma 0.3333333333333333 (* t_m t_m) 1.0) l) k)
          k
          (* (/ (* t_m t_m) l) 2.0))
         (* k k))))
      (if (<= k 1.4e+154)
        (/ 2.0 (* (* (/ (* k k) l) t_m) (* (/ (sin k) l) (tan k))))
        (/ 2.0 (* (* (/ (* (* (sin k) (tan k)) (/ t_m l)) l) 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 <= 1.5e-83) {
		tmp = (((2.0 / t_m) * l) / (2.0 * k)) / ((tan(k) * t_m) * (t_m / l));
	} else if (k <= 0.0275) {
		tmp = 2.0 / ((t_m / l) * (fma(((fma(0.3333333333333333, (t_m * t_m), 1.0) / l) * k), k, (((t_m * t_m) / l) * 2.0)) * (k * k)));
	} else if (k <= 1.4e+154) {
		tmp = 2.0 / ((((k * k) / l) * t_m) * ((sin(k) / l) * tan(k)));
	} else {
		tmp = 2.0 / (((((sin(k) * tan(k)) * (t_m / l)) / l) * 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 <= 1.5e-83)
		tmp = Float64(Float64(Float64(Float64(2.0 / t_m) * l) / Float64(2.0 * k)) / Float64(Float64(tan(k) * t_m) * Float64(t_m / l)));
	elseif (k <= 0.0275)
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(fma(Float64(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) / l) * k), k, Float64(Float64(Float64(t_m * t_m) / l) * 2.0)) * Float64(k * k))));
	elseif (k <= 1.4e+154)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) / l) * t_m) * Float64(Float64(sin(k) / l) * tan(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(sin(k) * tan(k)) * Float64(t_m / l)) / l) * k) * k));
	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[k, 1.5e-83], N[(N[(N[(N[(2.0 / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(2.0 * k), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.0275], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.4e+154], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if k < 1.50000000000000005e-83

   1. Initial program 62.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 65.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. associate-+l+ N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

   6. Applied rewrites 79.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 85.6%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 81.3

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

   11. Applied rewrites 81.3%

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

   if 1.50000000000000005e-83 < k < 0.0275000000000000001

   1. Initial program 59.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 78.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. associate-+l+ N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

   6. Applied rewrites 82.2%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 99.6%

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

   if 0.0275000000000000001 < k < 1.4e154

   1. Initial program 57.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 81.5

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

   5. Applied rewrites 81.5%

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

   7. Applied rewrites 81.6%

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

   9. Applied rewrites 81.6%

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

   if 1.4e154 < k

   1. Initial program 58.2%

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

   3. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 84.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{\frac{{\sin k}^{2}}{\ell}}{\ell \cdot \cos k} \cdot \mathsf{fma}\left(2, \frac{\frac{{t}^{3}}{k}}{k}, t\right)\right) \cdot k\right) \cdot k}} \]

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 82.0%

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

   10. Applied rewrites 84.6%

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

Alternative 5: 79.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m} \cdot \ell}{2 \cdot k}}{\left(\tan k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{elif}\;k \leq 0.0275:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right)}{\ell} \cdot k, k, \frac{t\_m \cdot t\_m}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \sin k\right) \cdot \left(\left(k \cdot t\_m\right) \cdot \frac{k}{\ell}\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.5e-83)
    (/ (/ (* (/ 2.0 t_m) l) (* 2.0 k)) (* (* (tan k) t_m) (/ t_m l)))
    (if (<= k 0.0275)
      (/
       2.0
       (*
        (/ t_m l)
        (*
         (fma
          (* (/ (fma 0.3333333333333333 (* t_m t_m) 1.0) l) k)
          k
          (* (/ (* t_m t_m) l) 2.0))
         (* k k))))
      (/ 2.0 (/ (* (* (tan k) (sin k)) (* (* k t_m) (/ k l))) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.5e-83) {
		tmp = (((2.0 / t_m) * l) / (2.0 * k)) / ((tan(k) * t_m) * (t_m / l));
	} else if (k <= 0.0275) {
		tmp = 2.0 / ((t_m / l) * (fma(((fma(0.3333333333333333, (t_m * t_m), 1.0) / l) * k), k, (((t_m * t_m) / l) * 2.0)) * (k * k)));
	} else {
		tmp = 2.0 / (((tan(k) * sin(k)) * ((k * t_m) * (k / l))) / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.5e-83)
		tmp = Float64(Float64(Float64(Float64(2.0 / t_m) * l) / Float64(2.0 * k)) / Float64(Float64(tan(k) * t_m) * Float64(t_m / l)));
	elseif (k <= 0.0275)
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(fma(Float64(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) / l) * k), k, Float64(Float64(Float64(t_m * t_m) / l) * 2.0)) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * sin(k)) * Float64(Float64(k * t_m) * Float64(k / l))) / l));
	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[k, 1.5e-83], N[(N[(N[(N[(2.0 / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(2.0 * k), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.0275], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.50000000000000005e-83

   1. Initial program 62.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 65.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. associate-+l+ N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

   6. Applied rewrites 79.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 85.6%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 81.3

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

   11. Applied rewrites 81.3%

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

   if 1.50000000000000005e-83 < k < 0.0275000000000000001

   1. Initial program 59.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 78.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. associate-+l+ N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

   6. Applied rewrites 82.2%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 99.6%

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

   if 0.0275000000000000001 < k

   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

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 76.5

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

   5. Applied rewrites 76.5%

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

   7. Applied rewrites 76.6%

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

   9. Applied rewrites 87.5%

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

Alternative 6: 79.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m} \cdot \ell}{2 \cdot k}}{\left(\tan k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{elif}\;k \leq 0.0275:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right)}{\ell} \cdot k, k, \frac{t\_m \cdot t\_m}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \sin k\right) \cdot \left(k \cdot \left(\frac{k}{\ell} \cdot t\_m\right)\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.5e-83)
    (/ (/ (* (/ 2.0 t_m) l) (* 2.0 k)) (* (* (tan k) t_m) (/ t_m l)))
    (if (<= k 0.0275)
      (/
       2.0
       (*
        (/ t_m l)
        (*
         (fma
          (* (/ (fma 0.3333333333333333 (* t_m t_m) 1.0) l) k)
          k
          (* (/ (* t_m t_m) l) 2.0))
         (* k k))))
      (/ 2.0 (/ (* (* (tan k) (sin k)) (* k (* (/ k l) t_m))) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.5e-83) {
		tmp = (((2.0 / t_m) * l) / (2.0 * k)) / ((tan(k) * t_m) * (t_m / l));
	} else if (k <= 0.0275) {
		tmp = 2.0 / ((t_m / l) * (fma(((fma(0.3333333333333333, (t_m * t_m), 1.0) / l) * k), k, (((t_m * t_m) / l) * 2.0)) * (k * k)));
	} else {
		tmp = 2.0 / (((tan(k) * sin(k)) * (k * ((k / l) * t_m))) / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.5e-83)
		tmp = Float64(Float64(Float64(Float64(2.0 / t_m) * l) / Float64(2.0 * k)) / Float64(Float64(tan(k) * t_m) * Float64(t_m / l)));
	elseif (k <= 0.0275)
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(fma(Float64(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) / l) * k), k, Float64(Float64(Float64(t_m * t_m) / l) * 2.0)) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * sin(k)) * Float64(k * Float64(Float64(k / l) * t_m))) / l));
	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[k, 1.5e-83], N[(N[(N[(N[(2.0 / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(2.0 * k), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.0275], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(k * N[(N[(k / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.50000000000000005e-83

   1. Initial program 62.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 65.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. associate-+l+ N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

   6. Applied rewrites 79.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 85.6%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 81.3

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

   11. Applied rewrites 81.3%

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

   if 1.50000000000000005e-83 < k < 0.0275000000000000001

   1. Initial program 59.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 78.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. associate-+l+ N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

   6. Applied rewrites 82.2%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 99.6%

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

   if 0.0275000000000000001 < k

   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

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 76.5

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

   5. Applied rewrites 76.5%

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

   7. Applied rewrites 76.6%

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

   9. Applied rewrites 87.5%

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

Alternative 7: 76.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m} \cdot \ell}{2 \cdot k}}{\left(\tan k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{elif}\;k \leq 0.0275:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right)}{\ell} \cdot k, k, \frac{t\_m \cdot t\_m}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(\sin k \cdot \tan k\right) \cdot \frac{t\_m}{\ell}}{\ell} \cdot k\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.5e-83)
    (/ (/ (* (/ 2.0 t_m) l) (* 2.0 k)) (* (* (tan k) t_m) (/ t_m l)))
    (if (<= k 0.0275)
      (/
       2.0
       (*
        (/ t_m l)
        (*
         (fma
          (* (/ (fma 0.3333333333333333 (* t_m t_m) 1.0) l) k)
          k
          (* (/ (* t_m t_m) l) 2.0))
         (* k k))))
      (/ 2.0 (* (* (/ (* (* (sin k) (tan k)) (/ t_m l)) l) 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 <= 1.5e-83) {
		tmp = (((2.0 / t_m) * l) / (2.0 * k)) / ((tan(k) * t_m) * (t_m / l));
	} else if (k <= 0.0275) {
		tmp = 2.0 / ((t_m / l) * (fma(((fma(0.3333333333333333, (t_m * t_m), 1.0) / l) * k), k, (((t_m * t_m) / l) * 2.0)) * (k * k)));
	} else {
		tmp = 2.0 / (((((sin(k) * tan(k)) * (t_m / l)) / l) * 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 <= 1.5e-83)
		tmp = Float64(Float64(Float64(Float64(2.0 / t_m) * l) / Float64(2.0 * k)) / Float64(Float64(tan(k) * t_m) * Float64(t_m / l)));
	elseif (k <= 0.0275)
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(fma(Float64(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) / l) * k), k, Float64(Float64(Float64(t_m * t_m) / l) * 2.0)) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(sin(k) * tan(k)) * Float64(t_m / l)) / l) * k) * k));
	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[k, 1.5e-83], N[(N[(N[(N[(2.0 / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(2.0 * k), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.0275], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.50000000000000005e-83

   1. Initial program 62.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 65.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. associate-+l+ N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

   6. Applied rewrites 79.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 85.6%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 81.3

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

   11. Applied rewrites 81.3%

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

   if 1.50000000000000005e-83 < k < 0.0275000000000000001

   1. Initial program 59.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 78.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. associate-+l+ N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

   6. Applied rewrites 82.2%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 99.6%

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

   if 0.0275000000000000001 < k

   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 k around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 78.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{\frac{{\sin k}^{2}}{\ell}}{\ell \cdot \cos k} \cdot \mathsf{fma}\left(2, \frac{\frac{{t}^{3}}{k}}{k}, t\right)\right) \cdot k\right) \cdot k}} \]

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 75.5%

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

   10. Applied rewrites 81.1%

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

Alternative 8: 76.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m} \cdot \ell}{2 \cdot k}}{\left(\tan k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{elif}\;k \leq 0.029:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right)}{\ell} \cdot k, k, \frac{t\_m \cdot t\_m}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell \cdot \ell} \cdot k\right) \cdot k\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.5e-83)
    (/ (/ (* (/ 2.0 t_m) l) (* 2.0 k)) (* (* (tan k) t_m) (/ t_m l)))
    (if (<= k 0.029)
      (/
       2.0
       (*
        (/ t_m l)
        (*
         (fma
          (* (/ (fma 0.3333333333333333 (* t_m t_m) 1.0) l) k)
          k
          (* (/ (* t_m t_m) l) 2.0))
         (* k k))))
      (/ 2.0 (* (* (* (/ t_m (* l l)) k) k) (* (tan k) (sin k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.5e-83) {
		tmp = (((2.0 / t_m) * l) / (2.0 * k)) / ((tan(k) * t_m) * (t_m / l));
	} else if (k <= 0.029) {
		tmp = 2.0 / ((t_m / l) * (fma(((fma(0.3333333333333333, (t_m * t_m), 1.0) / l) * k), k, (((t_m * t_m) / l) * 2.0)) * (k * k)));
	} else {
		tmp = 2.0 / ((((t_m / (l * l)) * k) * k) * (tan(k) * sin(k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.5e-83)
		tmp = Float64(Float64(Float64(Float64(2.0 / t_m) * l) / Float64(2.0 * k)) / Float64(Float64(tan(k) * t_m) * Float64(t_m / l)));
	elseif (k <= 0.029)
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(fma(Float64(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) / l) * k), k, Float64(Float64(Float64(t_m * t_m) / l) * 2.0)) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / Float64(l * l)) * k) * k) * Float64(tan(k) * sin(k))));
	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[k, 1.5e-83], N[(N[(N[(N[(2.0 / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(2.0 * k), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.029], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.50000000000000005e-83

   1. Initial program 62.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 65.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. associate-+l+ N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

   6. Applied rewrites 79.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 85.6%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 81.3

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

   11. Applied rewrites 81.3%

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

   if 1.50000000000000005e-83 < k < 0.0290000000000000015

   1. Initial program 59.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 78.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. metadata-eval N/A

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

      16. associate-+l+ N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-+.f64 N/A

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

   6. Applied rewrites 82.2%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 99.6%

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

   if 0.0290000000000000015 < k

   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

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 76.5

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

   5. Applied rewrites 76.5%

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

   7. Applied rewrites 76.6%

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

   9. Applied rewrites 75.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot k\right) \cdot k\right) \cdot \left(\tan k \cdot \sin k\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 76.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m} \cdot \ell}{2 \cdot k}}{\left(\tan k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{elif}\;k \leq 0.029:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right)}{\ell} \cdot k, k, \frac{t\_m \cdot t\_m}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\sin k \cdot \left(\left(\frac{t\_m}{\ell \cdot \ell} \cdot k\right) \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 1.5e-83)
    (/ (/ (* (/ 2.0 t_m) l) (* 2.0 k)) (* (* (tan k) t_m) (/ t_m l)))
    (if (<= k 0.029)
      (/
       2.0
       (*
        (/ t_m l)
        (*
         (fma
          (* (/ (fma 0.3333333333333333 (* t_m t_m) 1.0) l) k)
          k
          (* (/ (* t_m t_m) l) 2.0))
         (* k k))))
      (/ 2.0 (* (tan k) (* (sin k) (* (* (/ t_m (* l l)) 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 <= 1.5e-83) {
		tmp = (((2.0 / t_m) * l) / (2.0 * k)) / ((tan(k) * t_m) * (t_m / l));
	} else if (k <= 0.029) {
		tmp = 2.0 / ((t_m / l) * (fma(((fma(0.3333333333333333, (t_m * t_m), 1.0) / l) * k), k, (((t_m * t_m) / l) * 2.0)) * (k * k)));
	} else {
		tmp = 2.0 / (tan(k) * (sin(k) * (((t_m / (l * l)) * 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 <= 1.5e-83)
		tmp = Float64(Float64(Float64(Float64(2.0 / t_m) * l) / Float64(2.0 * k)) / Float64(Float64(tan(k) * t_m) * Float64(t_m / l)));
	elseif (k <= 0.029)
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(fma(Float64(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) / l) * k), k, Float64(Float64(Float64(t_m * t_m) / l) * 2.0)) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(tan(k) * Float64(sin(k) * Float64(Float64(Float64(t_m / Float64(l * l)) * k) * k))));
	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[k, 1.5e-83], N[(N[(N[(N[(2.0 / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(2.0 * k), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.029], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.50000000000000005e-83

   1. Initial program 62.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      8. cube-mult N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell}} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

   4. Applied rewrites 65.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\color{blue}{\left(\tan k \cdot \sin k\right)} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \tan k\right)} \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(t \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 2\right)}\right)\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{\left(1 + 1\right)}\right)\right)\right)} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\right)\right)} \]

      17. +-commutative N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right)\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right)\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right)\right)} \]

   6. Applied rewrites 79.4%

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right)}} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{t}{\ell}}}{\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{t}{\ell}}}{\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k}} \]

   8. Applied rewrites 85.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t} \cdot \ell}{\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k}}{\left(\tan k \cdot t\right) \cdot \frac{t}{\ell}}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \ell}{\color{blue}{2 \cdot k}}}{\left(\tan k \cdot t\right) \cdot \frac{t}{\ell}} \]
   10. Step-by-step derivation

       1. lower-*.f64 81.3

          \[\leadsto \frac{\frac{\frac{2}{t} \cdot \ell}{\color{blue}{2 \cdot k}}}{\left(\tan k \cdot t\right) \cdot \frac{t}{\ell}} \]

   11. Applied rewrites 81.3%

       \[\leadsto \frac{\frac{\frac{2}{t} \cdot \ell}{\color{blue}{2 \cdot k}}}{\left(\tan k \cdot t\right) \cdot \frac{t}{\ell}} \]

   if 1.50000000000000005e-83 < k < 0.0290000000000000015

   1. Initial program 59.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      8. cube-mult N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell}} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

   4. Applied rewrites 78.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\color{blue}{\left(\tan k \cdot \sin k\right)} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \tan k\right)} \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(t \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 2\right)}\right)\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{\left(1 + 1\right)}\right)\right)\right)} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\right)\right)} \]

      17. +-commutative N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right)\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right)\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right)\right)} \]

   6. Applied rewrites 82.2%

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}\right)}} \]

   9. Applied rewrites 99.6%

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)}{\ell} \cdot k, k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right)}} \]

   if 0.0290000000000000015 < k

   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

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      2. times-frac N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{2}}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\frac{{k}^{2}}{\ell}}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{\color{blue}{k \cdot k}}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      16. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\cos k}} \]

      17. lower-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\cos k}} \]

      18. lower-cos.f64 76.5

          \[\leadsto \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\color{blue}{\cos k}}} \]

   5. Applied rewrites 76.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 76.6%

      \[\leadsto \frac{2}{\frac{\left(\tan k \cdot \sin k\right) \cdot \left(\frac{k \cdot k}{\ell} \cdot t\right)}{\color{blue}{\ell}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 75.6%

      \[\leadsto \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(\left(\frac{t}{\ell \cdot \ell} \cdot k\right) \cdot k\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 72.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_m} \cdot \ell}{2 \cdot k}}{\left(\tan k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{elif}\;k \leq 1.56:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right)}{\ell} \cdot k, k, \frac{t\_m \cdot t\_m}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(k \cdot k\right) \cdot t\_m}{\left(\cos k \cdot \ell\right) \cdot \ell} \cdot k\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.5e-83)
    (/ (/ (* (/ 2.0 t_m) l) (* 2.0 k)) (* (* (tan k) t_m) (/ t_m l)))
    (if (<= k 1.56)
      (/
       2.0
       (*
        (/ t_m l)
        (*
         (fma
          (* (/ (fma 0.3333333333333333 (* t_m t_m) 1.0) l) k)
          k
          (* (/ (* t_m t_m) l) 2.0))
         (* k k))))
      (/ 2.0 (* (* (/ (* (* k k) t_m) (* (* (cos k) l) l)) 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 <= 1.5e-83) {
		tmp = (((2.0 / t_m) * l) / (2.0 * k)) / ((tan(k) * t_m) * (t_m / l));
	} else if (k <= 1.56) {
		tmp = 2.0 / ((t_m / l) * (fma(((fma(0.3333333333333333, (t_m * t_m), 1.0) / l) * k), k, (((t_m * t_m) / l) * 2.0)) * (k * k)));
	} else {
		tmp = 2.0 / (((((k * k) * t_m) / ((cos(k) * l) * l)) * 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 <= 1.5e-83)
		tmp = Float64(Float64(Float64(Float64(2.0 / t_m) * l) / Float64(2.0 * k)) / Float64(Float64(tan(k) * t_m) * Float64(t_m / l)));
	elseif (k <= 1.56)
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(fma(Float64(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) / l) * k), k, Float64(Float64(Float64(t_m * t_m) / l) * 2.0)) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * k) * t_m) / Float64(Float64(cos(k) * l) * l)) * k) * k));
	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[k, 1.5e-83], N[(N[(N[(N[(2.0 / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(2.0 * k), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.56], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.50000000000000005e-83

   1. Initial program 62.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      8. cube-mult N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell}} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

   4. Applied rewrites 65.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\color{blue}{\left(\tan k \cdot \sin k\right)} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \tan k\right)} \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(t \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 2\right)}\right)\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{\left(1 + 1\right)}\right)\right)\right)} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\right)\right)} \]

      17. +-commutative N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right)\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right)\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right)\right)} \]

   6. Applied rewrites 79.4%

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right)}} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{t}{\ell}}}{\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{t}{\ell}}}{\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k}} \]

   8. Applied rewrites 85.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t} \cdot \ell}{\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k}}{\left(\tan k \cdot t\right) \cdot \frac{t}{\ell}}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \ell}{\color{blue}{2 \cdot k}}}{\left(\tan k \cdot t\right) \cdot \frac{t}{\ell}} \]
   10. Step-by-step derivation

       1. lower-*.f64 81.3

          \[\leadsto \frac{\frac{\frac{2}{t} \cdot \ell}{\color{blue}{2 \cdot k}}}{\left(\tan k \cdot t\right) \cdot \frac{t}{\ell}} \]

   11. Applied rewrites 81.3%

       \[\leadsto \frac{\frac{\frac{2}{t} \cdot \ell}{\color{blue}{2 \cdot k}}}{\left(\tan k \cdot t\right) \cdot \frac{t}{\ell}} \]

   if 1.50000000000000005e-83 < k < 1.5600000000000001

   1. Initial program 59.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      8. cube-mult N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell}} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

   4. Applied rewrites 78.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\color{blue}{\left(\tan k \cdot \sin k\right)} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \tan k\right)} \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(t \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 2\right)}\right)\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{\left(1 + 1\right)}\right)\right)\right)} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\right)\right)} \]

      17. +-commutative N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right)\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right)\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right)\right)} \]

   6. Applied rewrites 82.2%

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}\right)}} \]

   9. Applied rewrites 99.6%

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)}{\ell} \cdot k, k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right)}} \]

   if 1.5600000000000001 < k

   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 k around inf

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot {k}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot k\right) \cdot k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot k\right) \cdot k}} \]

   5. Applied rewrites 78.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{\frac{{\sin k}^{2}}{\ell}}{\ell \cdot \cos k} \cdot \mathsf{fma}\left(2, \frac{\frac{{t}^{3}}{k}}{k}, t\right)\right) \cdot k\right) \cdot k}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot k\right) \cdot k} \]
   7. Step-by-step derivation

   8. Applied rewrites 75.5%

      \[\leadsto \frac{2}{\left(\frac{{\sin k}^{2} \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell} \cdot k\right) \cdot k} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell} \cdot k\right) \cdot k} \]
   10. Step-by-step derivation

   11. Applied rewrites 63.3%

       \[\leadsto \frac{2}{\left(\frac{\left(k \cdot k\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell} \cdot k\right) \cdot k} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 72.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t\_m}}{\left(\tan k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{elif}\;k \leq 1.56:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right)}{\ell} \cdot k, k, \frac{t\_m \cdot t\_m}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(k \cdot k\right) \cdot t\_m}{\left(\cos k \cdot \ell\right) \cdot \ell} \cdot k\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.5e-83)
    (/ (/ l (* k t_m)) (* (* (tan k) t_m) (/ t_m l)))
    (if (<= k 1.56)
      (/
       2.0
       (*
        (/ t_m l)
        (*
         (fma
          (* (/ (fma 0.3333333333333333 (* t_m t_m) 1.0) l) k)
          k
          (* (/ (* t_m t_m) l) 2.0))
         (* k k))))
      (/ 2.0 (* (* (/ (* (* k k) t_m) (* (* (cos k) l) l)) 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 <= 1.5e-83) {
		tmp = (l / (k * t_m)) / ((tan(k) * t_m) * (t_m / l));
	} else if (k <= 1.56) {
		tmp = 2.0 / ((t_m / l) * (fma(((fma(0.3333333333333333, (t_m * t_m), 1.0) / l) * k), k, (((t_m * t_m) / l) * 2.0)) * (k * k)));
	} else {
		tmp = 2.0 / (((((k * k) * t_m) / ((cos(k) * l) * l)) * 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 <= 1.5e-83)
		tmp = Float64(Float64(l / Float64(k * t_m)) / Float64(Float64(tan(k) * t_m) * Float64(t_m / l)));
	elseif (k <= 1.56)
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(fma(Float64(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) / l) * k), k, Float64(Float64(Float64(t_m * t_m) / l) * 2.0)) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * k) * t_m) / Float64(Float64(cos(k) * l) * l)) * k) * k));
	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[k, 1.5e-83], N[(N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.56], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.50000000000000005e-83

   1. Initial program 62.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      8. cube-mult N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell}} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

   4. Applied rewrites 65.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\color{blue}{\left(\tan k \cdot \sin k\right)} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \tan k\right)} \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(t \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 2\right)}\right)\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{\left(1 + 1\right)}\right)\right)\right)} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\right)\right)} \]

      17. +-commutative N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right)\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right)\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right)\right)} \]

   6. Applied rewrites 79.4%

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right)}} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{t}{\ell}}}{\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{t}{\ell}}}{\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k}} \]

   8. Applied rewrites 85.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t} \cdot \ell}{\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k}}{\left(\tan k \cdot t\right) \cdot \frac{t}{\ell}}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\color{blue}{\frac{\ell}{k \cdot t}}}{\left(\tan k \cdot t\right) \cdot \frac{t}{\ell}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\ell}{k \cdot t}}}{\left(\tan k \cdot t\right) \cdot \frac{t}{\ell}} \]

       2. lower-*.f64 81.3

          \[\leadsto \frac{\frac{\ell}{\color{blue}{k \cdot t}}}{\left(\tan k \cdot t\right) \cdot \frac{t}{\ell}} \]

   11. Applied rewrites 81.3%

       \[\leadsto \frac{\color{blue}{\frac{\ell}{k \cdot t}}}{\left(\tan k \cdot t\right) \cdot \frac{t}{\ell}} \]

   if 1.50000000000000005e-83 < k < 1.5600000000000001

   1. Initial program 59.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      8. cube-mult N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell}} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

   4. Applied rewrites 78.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\color{blue}{\left(\tan k \cdot \sin k\right)} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \tan k\right)} \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(t \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 2\right)}\right)\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{\left(1 + 1\right)}\right)\right)\right)} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\right)\right)} \]

      17. +-commutative N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right)\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right)\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right)\right)} \]

   6. Applied rewrites 82.2%

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}\right)}} \]

   9. Applied rewrites 99.6%

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)}{\ell} \cdot k, k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right)}} \]

   if 1.5600000000000001 < k

   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 k around inf

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot {k}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot k\right) \cdot k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot k\right) \cdot k}} \]

   5. Applied rewrites 78.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{\frac{{\sin k}^{2}}{\ell}}{\ell \cdot \cos k} \cdot \mathsf{fma}\left(2, \frac{\frac{{t}^{3}}{k}}{k}, t\right)\right) \cdot k\right) \cdot k}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot k\right) \cdot k} \]
   7. Step-by-step derivation

   8. Applied rewrites 75.5%

      \[\leadsto \frac{2}{\left(\frac{{\sin k}^{2} \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell} \cdot k\right) \cdot k} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell} \cdot k\right) \cdot k} \]
   10. Step-by-step derivation

   11. Applied rewrites 63.3%

       \[\leadsto \frac{2}{\left(\frac{\left(k \cdot k\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell} \cdot k\right) \cdot k} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 72.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t\_m}}{\left(\tan k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right)}{\ell} \cdot k, k, \frac{t\_m \cdot t\_m}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 1.5e-83)
    (/ (/ l (* k t_m)) (* (* (tan k) t_m) (/ t_m l)))
    (/
     2.0
     (*
      (/ t_m l)
      (*
       (fma
        (* (/ (fma 0.3333333333333333 (* t_m t_m) 1.0) l) k)
        k
        (* (/ (* t_m t_m) l) 2.0))
       (* 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 <= 1.5e-83) {
		tmp = (l / (k * t_m)) / ((tan(k) * t_m) * (t_m / l));
	} else {
		tmp = 2.0 / ((t_m / l) * (fma(((fma(0.3333333333333333, (t_m * t_m), 1.0) / l) * k), k, (((t_m * t_m) / l) * 2.0)) * (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 <= 1.5e-83)
		tmp = Float64(Float64(l / Float64(k * t_m)) / Float64(Float64(tan(k) * t_m) * Float64(t_m / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(fma(Float64(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) / l) * k), k, Float64(Float64(Float64(t_m * t_m) / l) * 2.0)) * Float64(k * k))));
	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[k, 1.5e-83], N[(N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.50000000000000005e-83

   1. Initial program 62.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      8. cube-mult N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell}} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

   4. Applied rewrites 65.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\color{blue}{\left(\tan k \cdot \sin k\right)} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \tan k\right)} \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(t \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 2\right)}\right)\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{\left(1 + 1\right)}\right)\right)\right)} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\right)\right)} \]

      17. +-commutative N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right)\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right)\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right)\right)} \]

   6. Applied rewrites 79.4%

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right)}} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{t}{\ell}}}{\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{t}{\ell}}}{\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k}} \]

   8. Applied rewrites 85.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t} \cdot \ell}{\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k}}{\left(\tan k \cdot t\right) \cdot \frac{t}{\ell}}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\color{blue}{\frac{\ell}{k \cdot t}}}{\left(\tan k \cdot t\right) \cdot \frac{t}{\ell}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\ell}{k \cdot t}}}{\left(\tan k \cdot t\right) \cdot \frac{t}{\ell}} \]

       2. lower-*.f64 81.3

          \[\leadsto \frac{\frac{\ell}{\color{blue}{k \cdot t}}}{\left(\tan k \cdot t\right) \cdot \frac{t}{\ell}} \]

   11. Applied rewrites 81.3%

       \[\leadsto \frac{\color{blue}{\frac{\ell}{k \cdot t}}}{\left(\tan k \cdot t\right) \cdot \frac{t}{\ell}} \]

   if 1.50000000000000005e-83 < k

   1. Initial program 58.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      8. cube-mult N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell}} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

   4. Applied rewrites 71.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\color{blue}{\left(\tan k \cdot \sin k\right)} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \tan k\right)} \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(t \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 2\right)}\right)\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{\left(1 + 1\right)}\right)\right)\right)} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\right)\right)} \]

      17. +-commutative N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right)\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right)\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right)\right)} \]

   6. Applied rewrites 76.1%

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}\right)}} \]

   9. Applied rewrites 68.8%

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)}{\ell} \cdot k, k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 73.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{\ell} \cdot k\right)}^{2} \cdot \left(t\_m \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right)}{\ell} \cdot k, k, \frac{t\_m \cdot t\_m}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 1.5e-83)
    (/ 2.0 (* (pow (* (/ t_m l) k) 2.0) (* t_m 2.0)))
    (/
     2.0
     (*
      (/ t_m l)
      (*
       (fma
        (* (/ (fma 0.3333333333333333 (* t_m t_m) 1.0) l) k)
        k
        (* (/ (* t_m t_m) l) 2.0))
       (* 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 <= 1.5e-83) {
		tmp = 2.0 / (pow(((t_m / l) * k), 2.0) * (t_m * 2.0));
	} else {
		tmp = 2.0 / ((t_m / l) * (fma(((fma(0.3333333333333333, (t_m * t_m), 1.0) / l) * k), k, (((t_m * t_m) / l) * 2.0)) * (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 <= 1.5e-83)
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / l) * k) ^ 2.0) * Float64(t_m * 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(fma(Float64(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) / l) * k), k, Float64(Float64(Float64(t_m * t_m) / l) * 2.0)) * Float64(k * k))));
	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[k, 1.5e-83], N[(2.0 / N[(N[Power[N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.50000000000000005e-83

   1. Initial program 62.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} \cdot 2}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot 2} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(\frac{{t}^{3}}{{\ell}^{2}} \cdot 2\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      11. associate-/r* N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      14. lower-pow.f64 63.2

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 63.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.4%

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{t \cdot \left(\frac{t}{\ell} \cdot t\right)}{\ell}} \]
   8. Step-by-step derivation

   9. Applied rewrites 68.4%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot t\right) \cdot {\left(\frac{\ell}{t}\right)}^{-2}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 82.2%

       \[\leadsto \color{blue}{\frac{2}{{\left(\frac{t}{\ell} \cdot k\right)}^{2} \cdot \left(t \cdot 2\right)}} \]

   if 1.50000000000000005e-83 < k

   1. Initial program 58.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      8. cube-mult N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell}} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

   4. Applied rewrites 71.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\color{blue}{\left(\tan k \cdot \sin k\right)} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \tan k\right)} \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(t \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 2\right)}\right)\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{\left(1 + 1\right)}\right)\right)\right)} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\right)\right)} \]

      17. +-commutative N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right)\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right)\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right)\right)} \]

   6. Applied rewrites 76.1%

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}\right)}} \]

   9. Applied rewrites 68.8%

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)}{\ell} \cdot k, k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 65.8% accurate, 4.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.4 \cdot 10^{-167}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{t\_m}{\ell} \cdot k}{\ell} \cdot k\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right)}{\ell} \cdot k, k, \frac{t\_m \cdot t\_m}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 5.4e-167)
    (/ 2.0 (* (* (* (/ (* (/ t_m l) k) l) k) k) k))
    (/
     2.0
     (*
      (/ t_m l)
      (*
       (fma
        (* (/ (fma 0.3333333333333333 (* t_m t_m) 1.0) l) k)
        k
        (* (/ (* t_m t_m) l) 2.0))
       (* 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 <= 5.4e-167) {
		tmp = 2.0 / ((((((t_m / l) * k) / l) * k) * k) * k);
	} else {
		tmp = 2.0 / ((t_m / l) * (fma(((fma(0.3333333333333333, (t_m * t_m), 1.0) / l) * k), k, (((t_m * t_m) / l) * 2.0)) * (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 <= 5.4e-167)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m / l) * k) / l) * k) * k) * k));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(fma(Float64(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) / l) * k), k, Float64(Float64(Float64(t_m * t_m) / l) * 2.0)) * Float64(k * k))));
	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[k, 5.4e-167], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.4000000000000001e-167

   1. Initial program 63.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around inf

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot {k}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot k\right) \cdot k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot k\right) \cdot k}} \]

   5. Applied rewrites 65.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{\frac{{\sin k}^{2}}{\ell}}{\ell \cdot \cos k} \cdot \mathsf{fma}\left(2, \frac{\frac{{t}^{3}}{k}}{k}, t\right)\right) \cdot k\right) \cdot k}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot k\right) \cdot k} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.7%

      \[\leadsto \frac{2}{\left(\frac{{\sin k}^{2} \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell} \cdot k\right) \cdot k} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot k\right) \cdot k} \]
   10. Step-by-step derivation

   11. Applied rewrites 64.0%

       \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell \cdot \ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot k} \]
   12. Step-by-step derivation

   13. Applied rewrites 69.9%

       \[\leadsto \frac{2}{\left(\left(\frac{\frac{t}{\ell} \cdot k}{\ell} \cdot k\right) \cdot k\right) \cdot k} \]

   if 5.4000000000000001e-167 < k

   1. Initial program 58.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      8. cube-mult N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell}} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)}} \]

   4. Applied rewrites 71.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\color{blue}{\left(\tan k \cdot \sin k\right)} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \tan k\right)} \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(t \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 2\right)}\right)\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{\left(1 + 1\right)}\right)\right)\right)} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\right)\right)} \]

      17. +-commutative N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right)\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)\right)\right)} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right)\right)} \]

   6. Applied rewrites 76.0%

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \left(\sin k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}\right)}} \]

   9. Applied rewrites 70.8%

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)}{\ell} \cdot k, k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 64.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 8.8 \cdot 10^{-94} \lor \neg \left(\ell \leq 1.1 \cdot 10^{+148}\right):\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{t\_m}{\ell} \cdot k}{\ell} \cdot k\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot t\_m\right) \cdot 2\right) \cdot t\_m}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (or (<= l 8.8e-94) (not (<= l 1.1e+148)))
    (/ 2.0 (* (* (* (/ (* (/ t_m l) k) l) k) k) k))
    (/ 2.0 (/ (* (* (* (* (* k k) t_m) t_m) 2.0) t_m) (* 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 ((l <= 8.8e-94) || !(l <= 1.1e+148)) {
		tmp = 2.0 / ((((((t_m / l) * k) / l) * k) * k) * k);
	} else {
		tmp = 2.0 / ((((((k * k) * t_m) * t_m) * 2.0) * t_m) / (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 ((l <= 8.8d-94) .or. (.not. (l <= 1.1d+148))) then
        tmp = 2.0d0 / ((((((t_m / l) * k) / l) * k) * k) * k)
    else
        tmp = 2.0d0 / ((((((k * k) * t_m) * t_m) * 2.0d0) * t_m) / (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 ((l <= 8.8e-94) || !(l <= 1.1e+148)) {
		tmp = 2.0 / ((((((t_m / l) * k) / l) * k) * k) * k);
	} else {
		tmp = 2.0 / ((((((k * k) * t_m) * t_m) * 2.0) * t_m) / (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 (l <= 8.8e-94) or not (l <= 1.1e+148):
		tmp = 2.0 / ((((((t_m / l) * k) / l) * k) * k) * k)
	else:
		tmp = 2.0 / ((((((k * k) * t_m) * t_m) * 2.0) * t_m) / (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 ((l <= 8.8e-94) || !(l <= 1.1e+148))
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m / l) * k) / l) * k) * k) * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(k * k) * t_m) * t_m) * 2.0) * t_m) / 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 ((l <= 8.8e-94) || ~((l <= 1.1e+148)))
		tmp = 2.0 / ((((((t_m / l) * k) / l) * k) * k) * k);
	else
		tmp = 2.0 / ((((((k * k) * t_m) * t_m) * 2.0) * t_m) / (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[Or[LessEqual[l, 8.8e-94], N[Not[LessEqual[l, 1.1e+148]], $MachinePrecision]], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 8.80000000000000004e-94 or 1.0999999999999999e148 < l

   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 k around inf

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot {k}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot k\right) \cdot k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot k\right) \cdot k}} \]

   5. Applied rewrites 69.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{\frac{{\sin k}^{2}}{\ell}}{\ell \cdot \cos k} \cdot \mathsf{fma}\left(2, \frac{\frac{{t}^{3}}{k}}{k}, t\right)\right) \cdot k\right) \cdot k}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot k\right) \cdot k} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.0%

      \[\leadsto \frac{2}{\left(\frac{{\sin k}^{2} \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell} \cdot k\right) \cdot k} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot k\right) \cdot k} \]
   10. Step-by-step derivation

   11. Applied rewrites 61.0%

       \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell \cdot \ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot k} \]
   12. Step-by-step derivation

   13. Applied rewrites 68.7%

       \[\leadsto \frac{2}{\left(\left(\frac{\frac{t}{\ell} \cdot k}{\ell} \cdot k\right) \cdot k\right) \cdot k} \]

   if 8.80000000000000004e-94 < l < 1.0999999999999999e148

   1. Initial program 72.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t} \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-log.f64 37.6

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 37.6%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 71.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(t \cdot t\right)\right) \cdot t}{\ell \cdot \ell}}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(2 \cdot \left({k}^{2} \cdot {t}^{2}\right)\right)} \cdot t}{\ell \cdot \ell}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left({k}^{2} \cdot {t}^{2}\right) \cdot 2\right)} \cdot t}{\ell \cdot \ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left({k}^{2} \cdot {t}^{2}\right) \cdot 2\right)} \cdot t}{\ell \cdot \ell}} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\left({k}^{2} \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot 2\right) \cdot t}{\ell \cdot \ell}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left(\color{blue}{\left(\left({k}^{2} \cdot t\right) \cdot t\right)} \cdot 2\right) \cdot t}{\ell \cdot \ell}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\color{blue}{\left(\left({k}^{2} \cdot t\right) \cdot t\right)} \cdot 2\right) \cdot t}{\ell \cdot \ell}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\color{blue}{\left({k}^{2} \cdot t\right)} \cdot t\right) \cdot 2\right) \cdot t}{\ell \cdot \ell}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot t\right) \cdot 2\right) \cdot t}{\ell \cdot \ell}} \]

      8. lower-*.f64 69.7

         \[\leadsto \frac{2}{\frac{\left(\left(\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot t\right) \cdot 2\right) \cdot t}{\ell \cdot \ell}} \]

   9. Applied rewrites 69.7%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot 2\right)} \cdot t}{\ell \cdot \ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.8 \cdot 10^{-94} \lor \neg \left(\ell \leq 1.1 \cdot 10^{+148}\right):\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{t}{\ell} \cdot k}{\ell} \cdot k\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot 2\right) \cdot t}{\ell \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 65.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 8.8 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{t\_m}{\ell} \cdot k}{\ell} \cdot k\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot t\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 8.8e-94)
    (/ 2.0 (* (* (* (/ (* (/ t_m l) k) l) k) k) k))
    (/ 2.0 (* (/ t_m l) (* (/ t_m l) (* (* (* k k) 2.0) t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 8.8e-94) {
		tmp = 2.0 / ((((((t_m / l) * k) / l) * k) * k) * k);
	} else {
		tmp = 2.0 / ((t_m / l) * ((t_m / l) * (((k * k) * 2.0) * t_m)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 8.8d-94) then
        tmp = 2.0d0 / ((((((t_m / l) * k) / l) * k) * k) * k)
    else
        tmp = 2.0d0 / ((t_m / l) * ((t_m / l) * (((k * k) * 2.0d0) * t_m)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 8.8e-94) {
		tmp = 2.0 / ((((((t_m / l) * k) / l) * k) * k) * k);
	} else {
		tmp = 2.0 / ((t_m / l) * ((t_m / l) * (((k * k) * 2.0) * t_m)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 8.8e-94:
		tmp = 2.0 / ((((((t_m / l) * k) / l) * k) * k) * k)
	else:
		tmp = 2.0 / ((t_m / l) * ((t_m / l) * (((k * k) * 2.0) * t_m)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 8.8e-94)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m / l) * k) / l) * k) * k) * k));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(t_m / l) * Float64(Float64(Float64(k * k) * 2.0) * t_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 8.8e-94)
		tmp = 2.0 / ((((((t_m / l) * k) / l) * k) * k) * k);
	else
		tmp = 2.0 / ((t_m / l) * ((t_m / l) * (((k * k) * 2.0) * t_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 8.8e-94], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 8.80000000000000004e-94

   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 inf

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot {k}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot k\right) \cdot k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot k\right) \cdot k}} \]

   5. Applied rewrites 72.4%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{\frac{{\sin k}^{2}}{\ell}}{\ell \cdot \cos k} \cdot \mathsf{fma}\left(2, \frac{\frac{{t}^{3}}{k}}{k}, t\right)\right) \cdot k\right) \cdot k}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot k\right) \cdot k} \]
   7. Step-by-step derivation

   8. Applied rewrites 64.7%

      \[\leadsto \frac{2}{\left(\frac{{\sin k}^{2} \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell} \cdot k\right) \cdot k} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot k\right) \cdot k} \]
   10. Step-by-step derivation

   11. Applied rewrites 60.6%

       \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell \cdot \ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot k} \]
   12. Step-by-step derivation

   13. Applied rewrites 69.1%

       \[\leadsto \frac{2}{\left(\left(\frac{\frac{t}{\ell} \cdot k}{\ell} \cdot k\right) \cdot k\right) \cdot k} \]

   if 8.80000000000000004e-94 < l

   1. Initial program 64.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} \cdot 2}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot 2} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(\frac{{t}^{3}}{{\ell}^{2}} \cdot 2\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      11. associate-/r* N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      14. lower-pow.f64 64.7

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 64.7%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 67.0%

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{t \cdot \left(\frac{t}{\ell} \cdot t\right)}{\ell}} \]
   8. Step-by-step derivation

   9. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot t\right) \cdot {\left(\frac{\ell}{t}\right)}^{-2}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 69.7%

       \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot t\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 64.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.55 \cdot 10^{-82}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{t\_m}{\ell} \cdot k}{\ell} \cdot k\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{t\_m \cdot \left(\frac{t\_m}{\ell} \cdot t\_m\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 1.55e-82)
    (/ 2.0 (* (* (* (/ (* (/ t_m l) k) l) k) k) k))
    (/ 2.0 (* (* (* k k) 2.0) (/ (* t_m (* (/ t_m l) t_m)) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 1.55e-82) {
		tmp = 2.0 / ((((((t_m / l) * k) / l) * k) * k) * k);
	} else {
		tmp = 2.0 / (((k * k) * 2.0) * ((t_m * ((t_m / l) * t_m)) / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 1.55d-82) then
        tmp = 2.0d0 / ((((((t_m / l) * k) / l) * k) * k) * k)
    else
        tmp = 2.0d0 / (((k * k) * 2.0d0) * ((t_m * ((t_m / l) * t_m)) / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 1.55e-82) {
		tmp = 2.0 / ((((((t_m / l) * k) / l) * k) * k) * k);
	} else {
		tmp = 2.0 / (((k * k) * 2.0) * ((t_m * ((t_m / l) * t_m)) / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 1.55e-82:
		tmp = 2.0 / ((((((t_m / l) * k) / l) * k) * k) * k)
	else:
		tmp = 2.0 / (((k * k) * 2.0) * ((t_m * ((t_m / l) * t_m)) / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 1.55e-82)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m / l) * k) / l) * k) * k) * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * 2.0) * Float64(Float64(t_m * Float64(Float64(t_m / l) * t_m)) / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 1.55e-82)
		tmp = 2.0 / ((((((t_m / l) * k) / l) * k) * k) * k);
	else
		tmp = 2.0 / (((k * k) * 2.0) * ((t_m * ((t_m / l) * t_m)) / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 1.55e-82], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.55e-82

   1. Initial program 60.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around inf

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot {k}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot k\right) \cdot k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot k\right) \cdot k}} \]

   5. Applied rewrites 72.7%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{\frac{{\sin k}^{2}}{\ell}}{\ell \cdot \cos k} \cdot \mathsf{fma}\left(2, \frac{\frac{{t}^{3}}{k}}{k}, t\right)\right) \cdot k\right) \cdot k}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot k\right) \cdot k} \]
   7. Step-by-step derivation

   8. Applied rewrites 65.1%

      \[\leadsto \frac{2}{\left(\frac{{\sin k}^{2} \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell} \cdot k\right) \cdot k} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot k\right) \cdot k} \]
   10. Step-by-step derivation

   11. Applied rewrites 61.1%

       \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell \cdot \ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot k} \]
   12. Step-by-step derivation

   13. Applied rewrites 69.4%

       \[\leadsto \frac{2}{\left(\left(\frac{\frac{t}{\ell} \cdot k}{\ell} \cdot k\right) \cdot k\right) \cdot k} \]

   if 1.55e-82 < l

   1. Initial program 63.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}} \cdot 2}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot 2} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(\frac{{t}^{3}}{{\ell}^{2}} \cdot 2\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot 2\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      11. associate-/r* N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      14. lower-pow.f64 63.9

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 63.9%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.2%

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{t \cdot \left(\frac{t}{\ell} \cdot t\right)}{\ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 64.3% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-188}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell \cdot \frac{\ell}{t\_m}} \cdot k\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot t\_m\right) \cdot 2\right) \cdot t\_m}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 5e-188)
    (/ 2.0 (* (* (* (/ k (* l (/ l t_m))) k) k) k))
    (/ 2.0 (/ (* (* (* (* (* k k) t_m) t_m) 2.0) t_m) (* 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 ((l * l) <= 5e-188) {
		tmp = 2.0 / ((((k / (l * (l / t_m))) * k) * k) * k);
	} else {
		tmp = 2.0 / ((((((k * k) * t_m) * t_m) * 2.0) * t_m) / (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 ((l * l) <= 5d-188) then
        tmp = 2.0d0 / ((((k / (l * (l / t_m))) * k) * k) * k)
    else
        tmp = 2.0d0 / ((((((k * k) * t_m) * t_m) * 2.0d0) * t_m) / (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 ((l * l) <= 5e-188) {
		tmp = 2.0 / ((((k / (l * (l / t_m))) * k) * k) * k);
	} else {
		tmp = 2.0 / ((((((k * k) * t_m) * t_m) * 2.0) * t_m) / (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 (l * l) <= 5e-188:
		tmp = 2.0 / ((((k / (l * (l / t_m))) * k) * k) * k)
	else:
		tmp = 2.0 / ((((((k * k) * t_m) * t_m) * 2.0) * t_m) / (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 (Float64(l * l) <= 5e-188)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / Float64(l * Float64(l / t_m))) * k) * k) * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(k * k) * t_m) * t_m) * 2.0) * t_m) / 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 ((l * l) <= 5e-188)
		tmp = 2.0 / ((((k / (l * (l / t_m))) * k) * k) * k);
	else
		tmp = 2.0 / ((((((k * k) * t_m) * t_m) * 2.0) * t_m) / (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[N[(l * l), $MachinePrecision], 5e-188], N[(2.0 / N[(N[(N[(N[(k / N[(l * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.0000000000000001e-188

   1. Initial program 61.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around inf

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot {k}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot k\right) \cdot k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot k\right) \cdot k}} \]

   5. Applied rewrites 79.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{\frac{{\sin k}^{2}}{\ell}}{\ell \cdot \cos k} \cdot \mathsf{fma}\left(2, \frac{\frac{{t}^{3}}{k}}{k}, t\right)\right) \cdot k\right) \cdot k}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot k\right) \cdot k} \]
   7. Step-by-step derivation

   8. Applied rewrites 65.5%

      \[\leadsto \frac{2}{\left(\frac{{\sin k}^{2} \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell} \cdot k\right) \cdot k} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot k\right) \cdot k} \]
   10. Step-by-step derivation

   11. Applied rewrites 70.9%

       \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell \cdot \ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot k} \]
   12. Step-by-step derivation

   13. Applied rewrites 79.1%

       \[\leadsto \frac{2}{\left(\left(\frac{k}{\ell \cdot \frac{\ell}{t}} \cdot k\right) \cdot k\right) \cdot k} \]

   if 5.0000000000000001e-188 < (*.f64 l l)

   1. Initial program 61.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t} \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-log.f64 19.4

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 19.4%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 59.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(t \cdot t\right)\right) \cdot t}{\ell \cdot \ell}}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(2 \cdot \left({k}^{2} \cdot {t}^{2}\right)\right)} \cdot t}{\ell \cdot \ell}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left({k}^{2} \cdot {t}^{2}\right) \cdot 2\right)} \cdot t}{\ell \cdot \ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left({k}^{2} \cdot {t}^{2}\right) \cdot 2\right)} \cdot t}{\ell \cdot \ell}} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\left({k}^{2} \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot 2\right) \cdot t}{\ell \cdot \ell}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left(\color{blue}{\left(\left({k}^{2} \cdot t\right) \cdot t\right)} \cdot 2\right) \cdot t}{\ell \cdot \ell}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\color{blue}{\left(\left({k}^{2} \cdot t\right) \cdot t\right)} \cdot 2\right) \cdot t}{\ell \cdot \ell}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\color{blue}{\left({k}^{2} \cdot t\right)} \cdot t\right) \cdot 2\right) \cdot t}{\ell \cdot \ell}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot t\right) \cdot 2\right) \cdot t}{\ell \cdot \ell}} \]

      8. lower-*.f64 60.4

         \[\leadsto \frac{2}{\frac{\left(\left(\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot t\right) \cdot 2\right) \cdot t}{\ell \cdot \ell}} \]

   9. Applied rewrites 60.4%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot 2\right)} \cdot t}{\ell \cdot \ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 64.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-188}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{\frac{t\_m}{\ell}}{\ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot t\_m\right) \cdot 2\right) \cdot t\_m}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 5e-188)
    (/ 2.0 (* (* (* (* (/ (/ t_m l) l) k) k) k) k))
    (/ 2.0 (/ (* (* (* (* (* k k) t_m) t_m) 2.0) t_m) (* 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 ((l * l) <= 5e-188) {
		tmp = 2.0 / ((((((t_m / l) / l) * k) * k) * k) * k);
	} else {
		tmp = 2.0 / ((((((k * k) * t_m) * t_m) * 2.0) * t_m) / (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 ((l * l) <= 5d-188) then
        tmp = 2.0d0 / ((((((t_m / l) / l) * k) * k) * k) * k)
    else
        tmp = 2.0d0 / ((((((k * k) * t_m) * t_m) * 2.0d0) * t_m) / (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 ((l * l) <= 5e-188) {
		tmp = 2.0 / ((((((t_m / l) / l) * k) * k) * k) * k);
	} else {
		tmp = 2.0 / ((((((k * k) * t_m) * t_m) * 2.0) * t_m) / (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 (l * l) <= 5e-188:
		tmp = 2.0 / ((((((t_m / l) / l) * k) * k) * k) * k)
	else:
		tmp = 2.0 / ((((((k * k) * t_m) * t_m) * 2.0) * t_m) / (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 (Float64(l * l) <= 5e-188)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m / l) / l) * k) * k) * k) * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(k * k) * t_m) * t_m) * 2.0) * t_m) / 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 ((l * l) <= 5e-188)
		tmp = 2.0 / ((((((t_m / l) / l) * k) * k) * k) * k);
	else
		tmp = 2.0 / ((((((k * k) * t_m) * t_m) * 2.0) * t_m) / (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[N[(l * l), $MachinePrecision], 5e-188], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.0000000000000001e-188

   1. Initial program 61.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around inf

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot {k}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot k\right) \cdot k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot k\right) \cdot k}} \]

   5. Applied rewrites 79.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{\frac{{\sin k}^{2}}{\ell}}{\ell \cdot \cos k} \cdot \mathsf{fma}\left(2, \frac{\frac{{t}^{3}}{k}}{k}, t\right)\right) \cdot k\right) \cdot k}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot k\right) \cdot k} \]
   7. Step-by-step derivation

   8. Applied rewrites 65.5%

      \[\leadsto \frac{2}{\left(\frac{{\sin k}^{2} \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell} \cdot k\right) \cdot k} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot k\right) \cdot k} \]
   10. Step-by-step derivation

   11. Applied rewrites 70.9%

       \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell \cdot \ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot k} \]
   12. Step-by-step derivation

   13. Applied rewrites 78.6%

       \[\leadsto \frac{2}{\left(\left(\left(\frac{\frac{t}{\ell}}{\ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot k} \]

   if 5.0000000000000001e-188 < (*.f64 l l)

   1. Initial program 61.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t} \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-log.f64 19.4

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 19.4%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 59.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(t \cdot t\right)\right) \cdot t}{\ell \cdot \ell}}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(2 \cdot \left({k}^{2} \cdot {t}^{2}\right)\right)} \cdot t}{\ell \cdot \ell}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left({k}^{2} \cdot {t}^{2}\right) \cdot 2\right)} \cdot t}{\ell \cdot \ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left({k}^{2} \cdot {t}^{2}\right) \cdot 2\right)} \cdot t}{\ell \cdot \ell}} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\left({k}^{2} \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot 2\right) \cdot t}{\ell \cdot \ell}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left(\color{blue}{\left(\left({k}^{2} \cdot t\right) \cdot t\right)} \cdot 2\right) \cdot t}{\ell \cdot \ell}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\color{blue}{\left(\left({k}^{2} \cdot t\right) \cdot t\right)} \cdot 2\right) \cdot t}{\ell \cdot \ell}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\color{blue}{\left({k}^{2} \cdot t\right)} \cdot t\right) \cdot 2\right) \cdot t}{\ell \cdot \ell}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot t\right) \cdot 2\right) \cdot t}{\ell \cdot \ell}} \]

      8. lower-*.f64 60.4

         \[\leadsto \frac{2}{\frac{\left(\left(\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot t\right) \cdot 2\right) \cdot t}{\ell \cdot \ell}} \]

   9. Applied rewrites 60.4%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot 2\right)} \cdot t}{\ell \cdot \ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 61.4% accurate, 7.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 1.8 \cdot 10^{-187}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m}{\ell \cdot \ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot t\_m\right) \cdot 2\right) \cdot t\_m}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 1.8e-187)
    (/ 2.0 (* (* (* (* (/ t_m (* l l)) k) k) k) k))
    (/ 2.0 (/ (* (* (* (* (* k k) t_m) t_m) 2.0) t_m) (* 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 ((l * l) <= 1.8e-187) {
		tmp = 2.0 / (((((t_m / (l * l)) * k) * k) * k) * k);
	} else {
		tmp = 2.0 / ((((((k * k) * t_m) * t_m) * 2.0) * t_m) / (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 ((l * l) <= 1.8d-187) then
        tmp = 2.0d0 / (((((t_m / (l * l)) * k) * k) * k) * k)
    else
        tmp = 2.0d0 / ((((((k * k) * t_m) * t_m) * 2.0d0) * t_m) / (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 ((l * l) <= 1.8e-187) {
		tmp = 2.0 / (((((t_m / (l * l)) * k) * k) * k) * k);
	} else {
		tmp = 2.0 / ((((((k * k) * t_m) * t_m) * 2.0) * t_m) / (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 (l * l) <= 1.8e-187:
		tmp = 2.0 / (((((t_m / (l * l)) * k) * k) * k) * k)
	else:
		tmp = 2.0 / ((((((k * k) * t_m) * t_m) * 2.0) * t_m) / (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 (Float64(l * l) <= 1.8e-187)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m / Float64(l * l)) * k) * k) * k) * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(k * k) * t_m) * t_m) * 2.0) * t_m) / 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 ((l * l) <= 1.8e-187)
		tmp = 2.0 / (((((t_m / (l * l)) * k) * k) * k) * k);
	else
		tmp = 2.0 / ((((((k * k) * t_m) * t_m) * 2.0) * t_m) / (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[N[(l * l), $MachinePrecision], 1.8e-187], N[(2.0 / N[(N[(N[(N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.79999999999999997e-187

   1. Initial program 61.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around inf

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot {k}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot k\right) \cdot k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot k\right) \cdot k}} \]

   5. Applied rewrites 79.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{\frac{{\sin k}^{2}}{\ell}}{\ell \cdot \cos k} \cdot \mathsf{fma}\left(2, \frac{\frac{{t}^{3}}{k}}{k}, t\right)\right) \cdot k\right) \cdot k}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot k\right) \cdot k} \]
   7. Step-by-step derivation

   8. Applied rewrites 65.5%

      \[\leadsto \frac{2}{\left(\frac{{\sin k}^{2} \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell} \cdot k\right) \cdot k} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot k\right) \cdot k} \]
   10. Step-by-step derivation

   11. Applied rewrites 70.9%

       \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell \cdot \ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot k} \]

   if 1.79999999999999997e-187 < (*.f64 l l)

   1. Initial program 61.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t} \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-log.f64 19.4

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 19.4%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 59.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(t \cdot t\right)\right) \cdot t}{\ell \cdot \ell}}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(2 \cdot \left({k}^{2} \cdot {t}^{2}\right)\right)} \cdot t}{\ell \cdot \ell}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left({k}^{2} \cdot {t}^{2}\right) \cdot 2\right)} \cdot t}{\ell \cdot \ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left({k}^{2} \cdot {t}^{2}\right) \cdot 2\right)} \cdot t}{\ell \cdot \ell}} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\left({k}^{2} \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot 2\right) \cdot t}{\ell \cdot \ell}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left(\color{blue}{\left(\left({k}^{2} \cdot t\right) \cdot t\right)} \cdot 2\right) \cdot t}{\ell \cdot \ell}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\color{blue}{\left(\left({k}^{2} \cdot t\right) \cdot t\right)} \cdot 2\right) \cdot t}{\ell \cdot \ell}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\color{blue}{\left({k}^{2} \cdot t\right)} \cdot t\right) \cdot 2\right) \cdot t}{\ell \cdot \ell}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot t\right) \cdot 2\right) \cdot t}{\ell \cdot \ell}} \]

      8. lower-*.f64 60.4

         \[\leadsto \frac{2}{\frac{\left(\left(\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot t\right) \cdot 2\right) \cdot t}{\ell \cdot \ell}} \]

   9. Applied rewrites 60.4%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot 2\right)} \cdot t}{\ell \cdot \ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 58.6% accurate, 9.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(\left(\left(\frac{t\_m}{\ell \cdot \ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot k} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* (* (* (/ t_m (* l l)) k) k) 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 * (2.0 / (((((t_m / (l * l)) * k) * k) * 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 * (2.0d0 / (((((t_m / (l * l)) * k) * k) * 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 * (2.0 / (((((t_m / (l * l)) * k) * k) * 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 * (2.0 / (((((t_m / (l * l)) * k) * k) * 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(2.0 / Float64(Float64(Float64(Float64(Float64(t_m / Float64(l * l)) * k) * k) * 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 * (2.0 / (((((t_m / (l * l)) * k) * k) * 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[(2.0 / N[(N[(N[(N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around inf

   \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot {k}^{2}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   3. associate-*r* N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot k\right) \cdot k}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot k\right) \cdot k}} \]

5. Applied rewrites 69.0%

   \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{\frac{{\sin k}^{2}}{\ell}}{\ell \cdot \cos k} \cdot \mathsf{fma}\left(2, \frac{\frac{{t}^{3}}{k}}{k}, t\right)\right) \cdot k\right) \cdot k}} \]

6. Taylor expanded in t around 0

   \[\leadsto \frac{2}{\left(\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot k\right) \cdot k} \]
7. Step-by-step derivation

8. Applied rewrites 66.8%

   \[\leadsto \frac{2}{\left(\frac{{\sin k}^{2} \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell} \cdot k\right) \cdot k} \]

9. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot k\right) \cdot k} \]
10. Step-by-step derivation

11. Applied rewrites 61.1%

    \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell \cdot \ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot k} \]
12. Add Preprocessing

Alternative 22: 57.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(\left(\left(t\_m \cdot \frac{k}{\ell \cdot \ell}\right) \cdot k\right) \cdot k\right) \cdot k} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* (* (* t_m (/ k (* l l))) k) 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 * (2.0 / ((((t_m * (k / (l * l))) * k) * 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 * (2.0d0 / ((((t_m * (k / (l * l))) * k) * 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 * (2.0 / ((((t_m * (k / (l * l))) * k) * 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 * (2.0 / ((((t_m * (k / (l * l))) * k) * 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(2.0 / Float64(Float64(Float64(Float64(t_m * Float64(k / Float64(l * l))) * k) * 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 * (2.0 / ((((t_m * (k / (l * l))) * k) * 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[(2.0 / N[(N[(N[(N[(t$95$m * N[(k / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around inf

   \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot {k}^{2}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   3. associate-*r* N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot k\right) \cdot k}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot k\right) \cdot k}} \]

5. Applied rewrites 69.0%

   \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{\frac{{\sin k}^{2}}{\ell}}{\ell \cdot \cos k} \cdot \mathsf{fma}\left(2, \frac{\frac{{t}^{3}}{k}}{k}, t\right)\right) \cdot k\right) \cdot k}} \]

6. Taylor expanded in t around 0

   \[\leadsto \frac{2}{\left(\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot k\right) \cdot k} \]
7. Step-by-step derivation

8. Applied rewrites 66.8%

   \[\leadsto \frac{2}{\left(\frac{{\sin k}^{2} \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell} \cdot k\right) \cdot k} \]

9. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot k\right) \cdot k} \]
10. Step-by-step derivation

11. Applied rewrites 61.1%

    \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell \cdot \ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot k} \]
12. Step-by-step derivation

13. Applied rewrites 59.9%

    \[\leadsto \frac{2}{\left(\left(\left(t \cdot \frac{k}{\ell \cdot \ell}\right) \cdot k\right) \cdot k\right) \cdot k} \]
14. Add Preprocessing

Alternative 23: 55.5% accurate, 9.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(\left(\frac{t\_m}{\ell \cdot \ell} \cdot k\right) \cdot k\right)} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* k k) (* (* (/ t_m (* l l)) 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 * (2.0 / ((k * k) * (((t_m / (l * l)) * 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 * (2.0d0 / ((k * k) * (((t_m / (l * l)) * 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 * (2.0 / ((k * k) * (((t_m / (l * l)) * 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 * (2.0 / ((k * k) * (((t_m / (l * l)) * 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(2.0 / Float64(Float64(k * k) * Float64(Float64(Float64(t_m / Float64(l * l)) * 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 * (2.0 / ((k * k) * (((t_m / (l * l)) * 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[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around inf

   \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot {k}^{2}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   3. associate-*r* N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot k\right) \cdot k}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\left(2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{k}^{2} \cdot \left({\ell}^{2} \cdot \cos k\right)} + \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot k\right) \cdot k}} \]

5. Applied rewrites 69.0%

   \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{\frac{{\sin k}^{2}}{\ell}}{\ell \cdot \cos k} \cdot \mathsf{fma}\left(2, \frac{\frac{{t}^{3}}{k}}{k}, t\right)\right) \cdot k\right) \cdot k}} \]

6. Taylor expanded in t around 0

   \[\leadsto \frac{2}{\left(\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot k\right) \cdot k} \]
7. Step-by-step derivation

8. Applied rewrites 66.8%

   \[\leadsto \frac{2}{\left(\frac{{\sin k}^{2} \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell} \cdot k\right) \cdot k} \]

9. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot k\right) \cdot k} \]
10. Step-by-step derivation

11. Applied rewrites 61.1%

    \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell \cdot \ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot k} \]
12. Step-by-step derivation

13. Applied rewrites 58.5%

    \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot \left(\left(\frac{t}{\ell \cdot \ell} \cdot k\right) \cdot k\right)}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024314 
(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))))