Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.1% → 88.2%

Time: 15.3s

Alternatives: 22

Speedup: 12.5×

• 27.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sin k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{+50}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(k \cdot \left(\tan k \cdot \frac{t\_2}{\ell}\right), k, \frac{\left(t\_m \cdot \left(t\_m \cdot 2\right)\right) \cdot \left(\tan k \cdot t\_2\right)}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \frac{t\_2 \cdot \left(\left(t\_m \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* t_m (sin k))))
   (*
    t_s
    (if (<= t_m 8e+50)
      (/
       2.0
       (/
        (fma
         (* k (* (tan k) (/ t_2 l)))
         k
         (/ (* (* t_m (* t_m 2.0)) (* (tan k) t_2)) l))
        l))
      (/
       2.0
       (*
        (/ t_m l)
        (/ (* t_2 (* (* t_m (tan k)) (fma k (/ k (* t_m t_m)) 2.0))) l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * sin(k);
	double tmp;
	if (t_m <= 8e+50) {
		tmp = 2.0 / (fma((k * (tan(k) * (t_2 / l))), k, (((t_m * (t_m * 2.0)) * (tan(k) * t_2)) / l)) / l);
	} else {
		tmp = 2.0 / ((t_m / l) * ((t_2 * ((t_m * tan(k)) * fma(k, (k / (t_m * t_m)), 2.0))) / l));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m * sin(k))
	tmp = 0.0
	if (t_m <= 8e+50)
		tmp = Float64(2.0 / Float64(fma(Float64(k * Float64(tan(k) * Float64(t_2 / l))), k, Float64(Float64(Float64(t_m * Float64(t_m * 2.0)) * Float64(tan(k) * t_2)) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(t_2 * Float64(Float64(t_m * tan(k)) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0))) / 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_] := Block[{t$95$2 = N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 8e+50], N[(2.0 / N[(N[(N[(k * N[(N[Tan[k], $MachinePrecision] * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k + N[(N[(N[(t$95$m * N[(t$95$m * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$2 * N[(N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.0000000000000006e50

   1. Initial program 50.7%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Simplified 70.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

   7. Applied egg-rr 82.4%

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

      1. lift-tan.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   9. Applied egg-rr 83.4%

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

       1. lift-tan.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. distribute-rgt-in N/A

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

       10. *-commutative N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*r* N/A

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

       13. lower-fma.f64 N/A

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

   11. Applied egg-rr 84.5%

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

   if 8.0000000000000006e50 < t

   1. Initial program 65.0%

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

   4. Applied egg-rr 64.4%

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

   6. Applied egg-rr 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*l/ N/A

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

   8. Applied egg-rr 97.6%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{+50}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(k \cdot \left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right), k, \frac{\left(t \cdot \left(t \cdot 2\right)\right) \cdot \left(\tan k \cdot \left(t \cdot \sin k\right)\right)}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{\left(t \cdot \sin k\right) \cdot \left(\left(t \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \tan k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.2 \cdot 10^{+50}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_2 \cdot \frac{\sin k}{\ell}\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t\_m \cdot t\_m\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \frac{\left(t\_m \cdot \sin k\right) \cdot \left(t\_2 \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* t_m (tan k))))
   (*
    t_s
    (if (<= t_m 8.2e+50)
      (/ 2.0 (/ (* (* t_2 (/ (sin k) l)) (fma k k (* 2.0 (* t_m t_m)))) l))
      (/
       2.0
       (*
        (/ t_m l)
        (/ (* (* t_m (sin k)) (* t_2 (fma k (/ k (* t_m t_m)) 2.0))) l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * tan(k);
	double tmp;
	if (t_m <= 8.2e+50) {
		tmp = 2.0 / (((t_2 * (sin(k) / l)) * fma(k, k, (2.0 * (t_m * t_m)))) / l);
	} else {
		tmp = 2.0 / ((t_m / l) * (((t_m * sin(k)) * (t_2 * fma(k, (k / (t_m * t_m)), 2.0))) / l));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m * tan(k))
	tmp = 0.0
	if (t_m <= 8.2e+50)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * Float64(sin(k) / l)) * fma(k, k, Float64(2.0 * Float64(t_m * t_m)))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(Float64(t_m * sin(k)) * Float64(t_2 * fma(k, Float64(k / Float64(t_m * t_m)), 2.0))) / 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_] := Block[{t$95$2 = N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 8.2e+50], N[(2.0 / N[(N[(N[(t$95$2 * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k * k + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.2000000000000002e50

   1. Initial program 50.7%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Simplified 70.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

   7. Applied egg-rr 82.4%

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

      1. lift-tan.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   9. Applied egg-rr 83.4%

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

   if 8.2000000000000002e50 < t

   1. Initial program 65.0%

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

   4. Applied egg-rr 64.4%

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

   6. Applied egg-rr 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*l/ N/A

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

   8. Applied egg-rr 97.6%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.2 \cdot 10^{+50}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(t \cdot \tan k\right) \cdot \frac{\sin k}{\ell}\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{\left(t \cdot \sin k\right) \cdot \left(\left(t \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sin k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.55 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \frac{t\_2}{\ell}}}{\frac{\mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \frac{\frac{t\_m}{\ell}}{\ell}\right) \cdot \left(t\_2 \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* t_m (sin k))))
   (*
    t_s
    (if (<= t_m 2.55e+160)
      (/ (/ 2.0 (* (tan k) (/ t_2 l))) (/ (fma 2.0 (* t_m t_m) (* k k)) l))
      (/
       2.0
       (*
        (* t_m (/ (/ t_m l) l))
        (* t_2 (* (tan k) (fma k (/ k (* t_m t_m)) 2.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * sin(k);
	double tmp;
	if (t_m <= 2.55e+160) {
		tmp = (2.0 / (tan(k) * (t_2 / l))) / (fma(2.0, (t_m * t_m), (k * k)) / l);
	} else {
		tmp = 2.0 / ((t_m * ((t_m / l) / l)) * (t_2 * (tan(k) * fma(k, (k / (t_m * t_m)), 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m * sin(k))
	tmp = 0.0
	if (t_m <= 2.55e+160)
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(t_2 / l))) / Float64(fma(2.0, Float64(t_m * t_m), Float64(k * k)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(t_m / l) / l)) * Float64(t_2 * Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.55e+160], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.5500000000000001e160

   1. Initial program 51.7%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Simplified 71.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

   7. Applied egg-rr 82.7%

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

      1. lift-tan.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   9. Applied egg-rr 84.0%

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

       1. lift-tan.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-fma.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. associate-/l* N/A

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

   11. Applied egg-rr 84.3%

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

   if 2.5500000000000001e160 < t

   1. Initial program 67.2%

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

   4. Applied egg-rr 54.5%

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

   6. Applied egg-rr 73.7%

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

      1. associate-/r* N/A

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

      2. lift-/.f64 N/A

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

      3. lower-/.f64 90.7

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

   8. Applied egg-rr 90.7%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.55 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \frac{t \cdot \sin k}{\ell}}}{\frac{\mathsf{fma}\left(2, t \cdot t, k \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{\frac{t}{\ell}}{\ell}\right) \cdot \left(\left(t \cdot \sin k\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(k, k, 2 \cdot \left(t\_m \cdot t\_m\right)\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-291}:\\ \;\;\;\;\frac{2}{\frac{t\_2 \cdot \left(k \cdot \left(k \cdot \mathsf{fma}\left(\frac{t\_m \cdot \left(k \cdot k\right)}{\ell}, 0.16666666666666666, \frac{t\_m}{\ell}\right)\right)\right)}{\ell}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+248}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \left(\tan k \cdot \left(t\_m \cdot \frac{\sin k}{\ell \cdot \ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(\left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right) \cdot \frac{2}{\ell}\right)}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (fma k k (* 2.0 (* t_m t_m)))))
   (*
    t_s
    (if (<= (* l l) 5e-291)
      (/
       2.0
       (/
        (*
         t_2
         (* k (* k (fma (/ (* t_m (* k k)) l) 0.16666666666666666 (/ t_m l)))))
        l))
      (if (<= (* l l) 5e+248)
        (/ 2.0 (* t_2 (* (tan k) (* t_m (/ (sin k) (* l l))))))
        (/ 2.0 (/ (* t_m (* (* k (* t_m (* t_m k))) (/ 2.0 l))) l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = fma(k, k, (2.0 * (t_m * t_m)));
	double tmp;
	if ((l * l) <= 5e-291) {
		tmp = 2.0 / ((t_2 * (k * (k * fma(((t_m * (k * k)) / l), 0.16666666666666666, (t_m / l))))) / l);
	} else if ((l * l) <= 5e+248) {
		tmp = 2.0 / (t_2 * (tan(k) * (t_m * (sin(k) / (l * l)))));
	} else {
		tmp = 2.0 / ((t_m * ((k * (t_m * (t_m * k))) * (2.0 / l))) / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = fma(k, k, Float64(2.0 * Float64(t_m * t_m)))
	tmp = 0.0
	if (Float64(l * l) <= 5e-291)
		tmp = Float64(2.0 / Float64(Float64(t_2 * Float64(k * Float64(k * fma(Float64(Float64(t_m * Float64(k * k)) / l), 0.16666666666666666, Float64(t_m / l))))) / l));
	elseif (Float64(l * l) <= 5e+248)
		tmp = Float64(2.0 / Float64(t_2 * Float64(tan(k) * Float64(t_m * Float64(sin(k) / Float64(l * l))))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(k * Float64(t_m * Float64(t_m * k))) * Float64(2.0 / 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_] := Block[{t$95$2 = N[(k * k + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 5e-291], N[(2.0 / N[(N[(t$95$2 * N[(k * N[(k * N[(N[(N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * 0.16666666666666666 + N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 5e+248], N[(2.0 / N[(t$95$2 * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[Sin[k], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[(N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 5.0000000000000003e-291

   1. Initial program 50.7%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Simplified 63.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

   7. Applied egg-rr 96.6%

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

      1. lift-tan.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   9. Applied egg-rr 96.9%

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

   10. Taylor expanded in k around 0

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

       1. unpow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell} + \frac{t}{\ell}\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       2. associate-*l* N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot \left(k \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell} + \frac{t}{\ell}\right)\right)\right)} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot \left(k \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell} + \frac{t}{\ell}\right)\right)\right)} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       4. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left(\color{blue}{\frac{{k}^{2} \cdot t}{\ell} \cdot \frac{1}{6}} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       5. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left(\color{blue}{\left({k}^{2} \cdot \frac{t}{\ell}\right)} \cdot \frac{1}{6} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       6. associate-*r* N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left(\color{blue}{{k}^{2} \cdot \left(\frac{t}{\ell} \cdot \frac{1}{6}\right)} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       7. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left({k}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{t}{\ell}\right)} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot \color{blue}{\left(k \cdot \left({k}^{2} \cdot \left(\frac{1}{6} \cdot \frac{t}{\ell}\right) + \frac{t}{\ell}\right)\right)}\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       9. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left({k}^{2} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{1}{6}\right)} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       10. associate-*r* N/A

           \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left(\color{blue}{\left({k}^{2} \cdot \frac{t}{\ell}\right) \cdot \frac{1}{6}} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       11. associate-/l* N/A

           \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left(\color{blue}{\frac{{k}^{2} \cdot t}{\ell}} \cdot \frac{1}{6} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       12. lower-fma.f64 N/A

           \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \color{blue}{\mathsf{fma}\left(\frac{{k}^{2} \cdot t}{\ell}, \frac{1}{6}, \frac{t}{\ell}\right)}\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

   12. Simplified 96.9%

       \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot \left(k \cdot \mathsf{fma}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}, 0.16666666666666666, \frac{t}{\ell}\right)\right)\right)} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

   if 5.0000000000000003e-291 < (*.f64 l l) < 4.9999999999999996e248

   1. Initial program 67.3%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Simplified 84.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   7. Applied egg-rr 88.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 88.5

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

   9. Applied egg-rr 88.5%

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

   if 4.9999999999999996e248 < (*.f64 l l)

   1. Initial program 35.0%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Simplified 53.9%

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

   6. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 55.4

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

   8. Simplified 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   10. Applied egg-rr 66.6%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-291}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot \left(k \cdot \mathsf{fma}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}, 0.16666666666666666, \frac{t}{\ell}\right)\right)\right)}{\ell}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+248}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right) \cdot \left(\tan k \cdot \left(t \cdot \frac{\sin k}{\ell \cdot \ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot \frac{2}{\ell}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(k, k, 2 \cdot \left(t\_m \cdot t\_m\right)\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-280}:\\ \;\;\;\;\frac{2}{\frac{t\_2 \cdot \left(k \cdot \left(k \cdot \mathsf{fma}\left(\frac{t\_m \cdot \left(k \cdot k\right)}{\ell}, 0.16666666666666666, \frac{t\_m}{\ell}\right)\right)\right)}{\ell}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+248}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \tan k\right) \cdot \left(t\_2 \cdot \frac{\sin k}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(\left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right) \cdot \frac{2}{\ell}\right)}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (fma k k (* 2.0 (* t_m t_m)))))
   (*
    t_s
    (if (<= (* l l) 5e-280)
      (/
       2.0
       (/
        (*
         t_2
         (* k (* k (fma (/ (* t_m (* k k)) l) 0.16666666666666666 (/ t_m l)))))
        l))
      (if (<= (* l l) 5e+248)
        (/ 2.0 (* (* t_m (tan k)) (* t_2 (/ (sin k) (* l l)))))
        (/ 2.0 (/ (* t_m (* (* k (* t_m (* t_m k))) (/ 2.0 l))) l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = fma(k, k, (2.0 * (t_m * t_m)));
	double tmp;
	if ((l * l) <= 5e-280) {
		tmp = 2.0 / ((t_2 * (k * (k * fma(((t_m * (k * k)) / l), 0.16666666666666666, (t_m / l))))) / l);
	} else if ((l * l) <= 5e+248) {
		tmp = 2.0 / ((t_m * tan(k)) * (t_2 * (sin(k) / (l * l))));
	} else {
		tmp = 2.0 / ((t_m * ((k * (t_m * (t_m * k))) * (2.0 / l))) / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = fma(k, k, Float64(2.0 * Float64(t_m * t_m)))
	tmp = 0.0
	if (Float64(l * l) <= 5e-280)
		tmp = Float64(2.0 / Float64(Float64(t_2 * Float64(k * Float64(k * fma(Float64(Float64(t_m * Float64(k * k)) / l), 0.16666666666666666, Float64(t_m / l))))) / l));
	elseif (Float64(l * l) <= 5e+248)
		tmp = Float64(2.0 / Float64(Float64(t_m * tan(k)) * Float64(t_2 * Float64(sin(k) / Float64(l * l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(k * Float64(t_m * Float64(t_m * k))) * Float64(2.0 / 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_] := Block[{t$95$2 = N[(k * k + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 5e-280], N[(2.0 / N[(N[(t$95$2 * N[(k * N[(k * N[(N[(N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * 0.16666666666666666 + N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 5e+248], N[(2.0 / N[(N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(N[Sin[k], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[(N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 5.00000000000000028e-280

   1. Initial program 49.9%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Simplified 64.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

   7. Applied egg-rr 96.7%

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

      1. lift-tan.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   9. Applied egg-rr 96.4%

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

   10. Taylor expanded in k around 0

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

       1. unpow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell} + \frac{t}{\ell}\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       2. associate-*l* N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot \left(k \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell} + \frac{t}{\ell}\right)\right)\right)} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot \left(k \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell} + \frac{t}{\ell}\right)\right)\right)} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       4. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left(\color{blue}{\frac{{k}^{2} \cdot t}{\ell} \cdot \frac{1}{6}} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       5. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left(\color{blue}{\left({k}^{2} \cdot \frac{t}{\ell}\right)} \cdot \frac{1}{6} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       6. associate-*r* N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left(\color{blue}{{k}^{2} \cdot \left(\frac{t}{\ell} \cdot \frac{1}{6}\right)} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       7. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left({k}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{t}{\ell}\right)} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot \color{blue}{\left(k \cdot \left({k}^{2} \cdot \left(\frac{1}{6} \cdot \frac{t}{\ell}\right) + \frac{t}{\ell}\right)\right)}\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       9. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left({k}^{2} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{1}{6}\right)} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       10. associate-*r* N/A

           \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left(\color{blue}{\left({k}^{2} \cdot \frac{t}{\ell}\right) \cdot \frac{1}{6}} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       11. associate-/l* N/A

           \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left(\color{blue}{\frac{{k}^{2} \cdot t}{\ell}} \cdot \frac{1}{6} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       12. lower-fma.f64 N/A

           \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \color{blue}{\mathsf{fma}\left(\frac{{k}^{2} \cdot t}{\ell}, \frac{1}{6}, \frac{t}{\ell}\right)}\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

   12. Simplified 96.6%

       \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot \left(k \cdot \mathsf{fma}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}, 0.16666666666666666, \frac{t}{\ell}\right)\right)\right)} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

   if 5.00000000000000028e-280 < (*.f64 l l) < 4.9999999999999996e248

   1. Initial program 68.2%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Simplified 84.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

   7. Applied egg-rr 85.8%

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

      1. lift-tan.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

   9. Applied egg-rr 87.9%

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

   if 4.9999999999999996e248 < (*.f64 l l)

   1. Initial program 35.0%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Simplified 53.9%

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

   6. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 55.4

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

   8. Simplified 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   10. Applied egg-rr 66.6%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-280}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot \left(k \cdot \mathsf{fma}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}, 0.16666666666666666, \frac{t}{\ell}\right)\right)\right)}{\ell}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+248}:\\ \;\;\;\;\frac{2}{\left(t \cdot \tan k\right) \cdot \left(\mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot \frac{2}{\ell}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.5 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \frac{t\_m \cdot \sin k}{\ell}}}{\frac{\mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(\left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right) \cdot \frac{2}{\ell}\right)}{\ell}}\\ \end{array} \end{array} \]

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 (<= t_m 7.5e+148)
    (/
     (/ 2.0 (* (tan k) (/ (* t_m (sin k)) l)))
     (/ (fma 2.0 (* t_m t_m) (* k k)) l))
    (/ 2.0 (/ (* t_m (* (* k (* t_m (* t_m k))) (/ 2.0 l))) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.5e+148) {
		tmp = (2.0 / (tan(k) * ((t_m * sin(k)) / l))) / (fma(2.0, (t_m * t_m), (k * k)) / l);
	} else {
		tmp = 2.0 / ((t_m * ((k * (t_m * (t_m * k))) * (2.0 / l))) / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.5e+148)
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(Float64(t_m * sin(k)) / l))) / Float64(fma(2.0, Float64(t_m * t_m), Float64(k * k)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(k * Float64(t_m * Float64(t_m * k))) * Float64(2.0 / 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[t$95$m, 7.5e+148], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[(N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.50000000000000008e148

   1. Initial program 51.9%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Simplified 71.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

   7. Applied egg-rr 83.0%

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

      1. lift-tan.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   9. Applied egg-rr 84.0%

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

       1. lift-tan.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-fma.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. associate-/l* N/A

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

   11. Applied egg-rr 84.6%

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

   if 7.50000000000000008e148 < t

   1. Initial program 65.4%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Simplified 62.1%

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

   6. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 68.4

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

   8. Simplified 68.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   10. Applied egg-rr 91.1%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \frac{t \cdot \sin k}{\ell}}}{\frac{\mathsf{fma}\left(2, t \cdot t, k \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot \frac{2}{\ell}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-135}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(\left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right) \cdot \frac{2}{\ell}\right)}{\ell}}\\ \mathbf{elif}\;k \leq 110:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(k, k, 2 \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \left(k \cdot \left(k \cdot \mathsf{fma}\left(\frac{t\_m \cdot \left(k \cdot k\right)}{\ell}, 0.16666666666666666, \frac{t\_m}{\ell}\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{\frac{\left(\tan k \cdot \left(t\_m \cdot \sin k\right)\right) \cdot \mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right)}{\ell}}\\ \end{array} \end{array} \]

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 1.35e-135)
    (/ 2.0 (/ (* t_m (* (* k (* t_m (* t_m k))) (/ 2.0 l))) l))
    (if (<= k 110.0)
      (/
       2.0
       (/
        (*
         (fma k k (* 2.0 (* t_m t_m)))
         (* k (* k (fma (/ (* t_m (* k k)) l) 0.16666666666666666 (/ t_m l)))))
        l))
      (/
       (* 2.0 l)
       (/ (* (* (tan k) (* t_m (sin k))) (fma 2.0 (* t_m t_m) (* k k))) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.35e-135) {
		tmp = 2.0 / ((t_m * ((k * (t_m * (t_m * k))) * (2.0 / l))) / l);
	} else if (k <= 110.0) {
		tmp = 2.0 / ((fma(k, k, (2.0 * (t_m * t_m))) * (k * (k * fma(((t_m * (k * k)) / l), 0.16666666666666666, (t_m / l))))) / l);
	} else {
		tmp = (2.0 * l) / (((tan(k) * (t_m * sin(k))) * fma(2.0, (t_m * t_m), (k * k))) / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.35e-135)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(k * Float64(t_m * Float64(t_m * k))) * Float64(2.0 / l))) / l));
	elseif (k <= 110.0)
		tmp = Float64(2.0 / Float64(Float64(fma(k, k, Float64(2.0 * Float64(t_m * t_m))) * Float64(k * Float64(k * fma(Float64(Float64(t_m * Float64(k * k)) / l), 0.16666666666666666, Float64(t_m / l))))) / l));
	else
		tmp = Float64(Float64(2.0 * l) / Float64(Float64(Float64(tan(k) * Float64(t_m * sin(k))) * fma(2.0, Float64(t_m * t_m), Float64(k * k))) / 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, 1.35e-135], N[(2.0 / N[(N[(t$95$m * N[(N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 110.0], N[(2.0 / N[(N[(N[(k * k + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k * N[(N[(N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * 0.16666666666666666 + N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * l), $MachinePrecision] / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.34999999999999999e-135

   1. Initial program 56.5%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Simplified 65.8%

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

   6. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 60.9

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

   8. Simplified 60.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   10. Applied egg-rr 73.1%

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

   if 1.34999999999999999e-135 < k < 110

   1. Initial program 44.0%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Simplified 70.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

   7. Applied egg-rr 91.3%

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

      1. lift-tan.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   9. Applied egg-rr 94.0%

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

   10. Taylor expanded in k around 0

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

       1. unpow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell} + \frac{t}{\ell}\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       2. associate-*l* N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot \left(k \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell} + \frac{t}{\ell}\right)\right)\right)} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot \left(k \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell} + \frac{t}{\ell}\right)\right)\right)} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       4. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left(\color{blue}{\frac{{k}^{2} \cdot t}{\ell} \cdot \frac{1}{6}} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       5. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left(\color{blue}{\left({k}^{2} \cdot \frac{t}{\ell}\right)} \cdot \frac{1}{6} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       6. associate-*r* N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left(\color{blue}{{k}^{2} \cdot \left(\frac{t}{\ell} \cdot \frac{1}{6}\right)} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       7. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left({k}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{t}{\ell}\right)} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot \color{blue}{\left(k \cdot \left({k}^{2} \cdot \left(\frac{1}{6} \cdot \frac{t}{\ell}\right) + \frac{t}{\ell}\right)\right)}\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       9. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left({k}^{2} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{1}{6}\right)} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       10. associate-*r* N/A

           \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left(\color{blue}{\left({k}^{2} \cdot \frac{t}{\ell}\right) \cdot \frac{1}{6}} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       11. associate-/l* N/A

           \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left(\color{blue}{\frac{{k}^{2} \cdot t}{\ell}} \cdot \frac{1}{6} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       12. lower-fma.f64 N/A

           \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \color{blue}{\mathsf{fma}\left(\frac{{k}^{2} \cdot t}{\ell}, \frac{1}{6}, \frac{t}{\ell}\right)}\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

   12. Simplified 96.6%

       \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot \left(k \cdot \mathsf{fma}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}, 0.16666666666666666, \frac{t}{\ell}\right)\right)\right)} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

   if 110 < k

   1. Initial program 52.5%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Simplified 79.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

   7. Applied egg-rr 81.3%

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

      1. lift-tan.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

   9. Applied egg-rr 79.8%

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

       1. lift-tan.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-fma.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. associate-/r/ N/A

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

       11. associate-*l/ N/A

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

       12. lower-/.f64 N/A

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

   11. Applied egg-rr 78.7%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-135}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot \frac{2}{\ell}\right)}{\ell}}\\ \mathbf{elif}\;k \leq 110:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot \left(k \cdot \mathsf{fma}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}, 0.16666666666666666, \frac{t}{\ell}\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{\frac{\left(\tan k \cdot \left(t \cdot \sin k\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.5 \cdot 10^{+148}:\\ \;\;\;\;\frac{2}{\left(\left(t\_m \cdot \tan k\right) \cdot \frac{\sin k}{\ell}\right) \cdot \frac{\mathsf{fma}\left(k, k, 2 \cdot \left(t\_m \cdot t\_m\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(\left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right) \cdot \frac{2}{\ell}\right)}{\ell}}\\ \end{array} \end{array} \]

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 (<= t_m 7.5e+148)
    (/
     2.0
     (* (* (* t_m (tan k)) (/ (sin k) l)) (/ (fma k k (* 2.0 (* t_m t_m))) l)))
    (/ 2.0 (/ (* t_m (* (* k (* t_m (* t_m k))) (/ 2.0 l))) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.5e+148) {
		tmp = 2.0 / (((t_m * tan(k)) * (sin(k) / l)) * (fma(k, k, (2.0 * (t_m * t_m))) / l));
	} else {
		tmp = 2.0 / ((t_m * ((k * (t_m * (t_m * k))) * (2.0 / l))) / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.5e+148)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * tan(k)) * Float64(sin(k) / l)) * Float64(fma(k, k, Float64(2.0 * Float64(t_m * t_m))) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(k * Float64(t_m * Float64(t_m * k))) * Float64(2.0 / 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[t$95$m, 7.5e+148], N[(2.0 / N[(N[(N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[(N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.50000000000000008e148

   1. Initial program 51.9%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Simplified 71.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

   7. Applied egg-rr 83.0%

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

      1. lift-tan.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

   9. Applied egg-rr 84.2%

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

   if 7.50000000000000008e148 < t

   1. Initial program 65.4%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Simplified 62.1%

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

   6. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 68.4

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

   8. Simplified 68.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   10. Applied egg-rr 91.1%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{+148}:\\ \;\;\;\;\frac{2}{\left(\left(t \cdot \tan k\right) \cdot \frac{\sin k}{\ell}\right) \cdot \frac{\mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot \frac{2}{\ell}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{+114}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left(\frac{\mathsf{fma}\left(k, k, 2 \cdot \left(t\_m \cdot t\_m\right)\right)}{\ell} \cdot \left(\tan k \cdot \frac{\sin k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(\left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right) \cdot \frac{2}{\ell}\right)}{\ell}}\\ \end{array} \end{array} \]

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 (<= t_m 4e+114)
    (/
     2.0
     (* t_m (* (/ (fma k k (* 2.0 (* t_m t_m))) l) (* (tan k) (/ (sin k) l)))))
    (/ 2.0 (/ (* t_m (* (* k (* t_m (* t_m k))) (/ 2.0 l))) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4e+114) {
		tmp = 2.0 / (t_m * ((fma(k, k, (2.0 * (t_m * t_m))) / l) * (tan(k) * (sin(k) / l))));
	} else {
		tmp = 2.0 / ((t_m * ((k * (t_m * (t_m * k))) * (2.0 / l))) / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4e+114)
		tmp = Float64(2.0 / Float64(t_m * Float64(Float64(fma(k, k, Float64(2.0 * Float64(t_m * t_m))) / l) * Float64(tan(k) * Float64(sin(k) / l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(k * Float64(t_m * Float64(t_m * k))) * Float64(2.0 / 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[t$95$m, 4e+114], N[(2.0 / N[(t$95$m * N[(N[(N[(k * k + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[(N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4e114

   1. Initial program 52.6%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Simplified 72.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

   7. Applied egg-rr 83.3%

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

   if 4e114 < t

   1. Initial program 59.4%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Simplified 61.6%

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

   6. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 66.9

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

   8. Simplified 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   10. Applied egg-rr 88.2%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{+114}:\\ \;\;\;\;\frac{2}{t \cdot \left(\frac{\mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell} \cdot \left(\tan k \cdot \frac{\sin k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot \frac{2}{\ell}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.9 \cdot 10^{+27}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \frac{k \cdot k}{\ell}\right)}\\ \mathbf{elif}\;t\_m \leq 9.4 \cdot 10^{+119}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \left(t\_m \cdot \left(\tan k \cdot \frac{\sin k}{\ell \cdot \ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(\left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right) \cdot \frac{2}{\ell}\right)}{\ell}}\\ \end{array} \end{array} \]

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 (<= t_m 1.9e+27)
    (/ 2.0 (* t_m (* (* (tan k) (/ (sin k) l)) (/ (* k k) l))))
    (if (<= t_m 9.4e+119)
      (/ 2.0 (* (* 2.0 (* t_m t_m)) (* t_m (* (tan k) (/ (sin k) (* l l))))))
      (/ 2.0 (/ (* t_m (* (* k (* t_m (* t_m k))) (/ 2.0 l))) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.9e+27) {
		tmp = 2.0 / (t_m * ((tan(k) * (sin(k) / l)) * ((k * k) / l)));
	} else if (t_m <= 9.4e+119) {
		tmp = 2.0 / ((2.0 * (t_m * t_m)) * (t_m * (tan(k) * (sin(k) / (l * l)))));
	} else {
		tmp = 2.0 / ((t_m * ((k * (t_m * (t_m * k))) * (2.0 / l))) / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.9d+27) then
        tmp = 2.0d0 / (t_m * ((tan(k) * (sin(k) / l)) * ((k * k) / l)))
    else if (t_m <= 9.4d+119) then
        tmp = 2.0d0 / ((2.0d0 * (t_m * t_m)) * (t_m * (tan(k) * (sin(k) / (l * l)))))
    else
        tmp = 2.0d0 / ((t_m * ((k * (t_m * (t_m * k))) * (2.0d0 / l))) / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.9e+27) {
		tmp = 2.0 / (t_m * ((Math.tan(k) * (Math.sin(k) / l)) * ((k * k) / l)));
	} else if (t_m <= 9.4e+119) {
		tmp = 2.0 / ((2.0 * (t_m * t_m)) * (t_m * (Math.tan(k) * (Math.sin(k) / (l * l)))));
	} else {
		tmp = 2.0 / ((t_m * ((k * (t_m * (t_m * k))) * (2.0 / l))) / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.9e+27:
		tmp = 2.0 / (t_m * ((math.tan(k) * (math.sin(k) / l)) * ((k * k) / l)))
	elif t_m <= 9.4e+119:
		tmp = 2.0 / ((2.0 * (t_m * t_m)) * (t_m * (math.tan(k) * (math.sin(k) / (l * l)))))
	else:
		tmp = 2.0 / ((t_m * ((k * (t_m * (t_m * k))) * (2.0 / l))) / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.9e+27)
		tmp = Float64(2.0 / Float64(t_m * Float64(Float64(tan(k) * Float64(sin(k) / l)) * Float64(Float64(k * k) / l))));
	elseif (t_m <= 9.4e+119)
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(t_m * t_m)) * Float64(t_m * Float64(tan(k) * Float64(sin(k) / Float64(l * l))))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(k * Float64(t_m * Float64(t_m * k))) * Float64(2.0 / 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 (t_m <= 1.9e+27)
		tmp = 2.0 / (t_m * ((tan(k) * (sin(k) / l)) * ((k * k) / l)));
	elseif (t_m <= 9.4e+119)
		tmp = 2.0 / ((2.0 * (t_m * t_m)) * (t_m * (tan(k) * (sin(k) / (l * l)))));
	else
		tmp = 2.0 / ((t_m * ((k * (t_m * (t_m * k))) * (2.0 / l))) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.9e+27], N[(2.0 / N[(t$95$m * N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.4e+119], N[(2.0 / N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[(N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.90000000000000011e27

   1. Initial program 49.7%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Simplified 70.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \left(2 \cdot \color{blue}{\left(t \cdot t\right)} + k \cdot k\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \left(2 \cdot \left(t \cdot t\right) + \color{blue}{k \cdot k}\right)\right)} \]

      8. lift-fma.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, k \cdot k\right)}\right)} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\frac{{\sin k}^{2} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{t \cdot \frac{{\sin k}^{2} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)}{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{t \cdot \frac{{\sin k}^{2} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)}{\color{blue}{\left(\ell \cdot \cos k\right) \cdot \ell}}} \]

      12. times-frac N/A

          \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{\ell \cdot \cos k} \cdot \frac{\mathsf{fma}\left(2, t \cdot t, k \cdot k\right)}{\ell}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{\ell \cdot \cos k} \cdot \frac{\mathsf{fma}\left(2, t \cdot t, k \cdot k\right)}{\ell}\right)}} \]

   7. Applied egg-rr 82.4%

      \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \frac{\mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \frac{2}{t \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \color{blue}{\frac{{k}^{2}}{\ell}}\right)} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \color{blue}{\frac{{k}^{2}}{\ell}}\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right)} \]

      3. lower-*.f64 72.8

         \[\leadsto \frac{2}{t \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right)} \]

   10. Simplified 72.8%

       \[\leadsto \frac{2}{t \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \color{blue}{\frac{k \cdot k}{\ell}}\right)} \]

   if 1.90000000000000011e27 < t < 9.40000000000000016e119

   1. Initial program 80.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2 \cdot \left({t}^{3} \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2 \cdot \left({t}^{3} \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(2 \cdot {t}^{3}\right) \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(2 \cdot {t}^{3}\right) \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      5. unpow3 N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot \left(\color{blue}{{t}^{2}} \cdot t\right)\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(2 \cdot {t}^{2}\right) \cdot t\right)} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(2 \cdot {t}^{2}\right) \cdot t\right)} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\color{blue}{\left(2 \cdot {t}^{2}\right)} \cdot t\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot \left(t \cdot t\right)\right) \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      13. lower-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot \left(t \cdot t\right)\right) \cdot t\right) \cdot {\color{blue}{\sin k}}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot \left(t \cdot t\right)\right) \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}} \]

      15. associate-*l* N/A

          \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot \left(t \cdot t\right)\right) \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot \left(t \cdot t\right)\right) \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot \left(t \cdot t\right)\right) \cdot t\right) \cdot {\sin k}^{2}}{\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}}} \]

      18. lower-cos.f64 85.4

          \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot \left(t \cdot t\right)\right) \cdot t\right) \cdot {\sin k}^{2}}{\ell \cdot \left(\ell \cdot \color{blue}{\cos k}\right)}} \]

   5. Simplified 85.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(2 \cdot \left(t \cdot t\right)\right) \cdot t\right) \cdot {\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t\right) \cdot {\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\color{blue}{\left(2 \cdot \left(t \cdot t\right)\right)} \cdot t\right) \cdot {\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(2 \cdot \left(t \cdot t\right)\right) \cdot t\right)} \cdot {\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)}} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot \left(t \cdot t\right)\right) \cdot t\right) \cdot {\color{blue}{\sin k}}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot \left(t \cdot t\right)\right) \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}}{\ell \cdot \left(\ell \cdot \cos k\right)}} \]

      6. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot \left(t \cdot t\right)\right) \cdot t\right) \cdot {\sin k}^{2}}{\ell \cdot \left(\ell \cdot \color{blue}{\cos k}\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot \left(t \cdot t\right)\right) \cdot t\right) \cdot {\sin k}^{2}}{\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot \left(t \cdot t\right)\right) \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

      9. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(2 \cdot \left(t \cdot t\right)\right) \cdot t\right) \cdot \frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\left(2 \cdot \left(t \cdot t\right)\right) \cdot t\right)} \cdot \frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)}} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(2 \cdot \left(t \cdot t\right)\right) \cdot t\right) \cdot \color{blue}{\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot \frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot \frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)}\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{2}{\left(2 \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot t\right)}} \]

      15. lower-*.f64 94.9

          \[\leadsto \frac{2}{\left(2 \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot t\right)}} \]

   7. Applied egg-rr 99.6%

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(t \cdot t\right)\right) \cdot \left(\left(\frac{\sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot t\right)}} \]

   if 9.40000000000000016e119 < t

   1. Initial program 59.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{t \cdot \left(2 \cdot \color{blue}{\left({t}^{2} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} \]

      4. associate-/l* N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(2 \cdot {t}^{2}\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \color{blue}{{k}^{2} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\right)} \]

      5. distribute-rgt-out N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \left(2 \cdot {t}^{2} + {k}^{2}\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \left(2 \cdot {t}^{2} + {k}^{2}\right)\right)}} \]

   5. Simplified 59.6%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{t \cdot \color{blue}{\left(2 \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\frac{2 \cdot \left({k}^{2} \cdot {t}^{2}\right)}{{\ell}^{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{t \cdot \frac{\color{blue}{\left({k}^{2} \cdot {t}^{2}\right) \cdot 2}}{{\ell}^{2}}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{t \cdot \frac{\color{blue}{{k}^{2} \cdot \left({t}^{2} \cdot 2\right)}}{{\ell}^{2}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{2} \cdot \color{blue}{\left(2 \cdot {t}^{2}\right)}}{{\ell}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\frac{{k}^{2} \cdot \left(2 \cdot {t}^{2}\right)}{{\ell}^{2}}}} \]

      6. *-commutative N/A

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{2} \cdot \color{blue}{\left({t}^{2} \cdot 2\right)}}{{\ell}^{2}}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{t \cdot \frac{\color{blue}{\left({k}^{2} \cdot {t}^{2}\right) \cdot 2}}{{\ell}^{2}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{t \cdot \frac{\color{blue}{2 \cdot \left({k}^{2} \cdot {t}^{2}\right)}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{\color{blue}{2 \cdot \left({k}^{2} \cdot {t}^{2}\right)}}{{\ell}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \color{blue}{\left({t}^{2} \cdot {k}^{2}\right)}}{{\ell}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot {k}^{2}\right)}{{\ell}^{2}}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \color{blue}{\left(t \cdot \left(t \cdot {k}^{2}\right)\right)}}{{\ell}^{2}}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)}{{\ell}^{2}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \color{blue}{\left(t \cdot \left({k}^{2} \cdot t\right)\right)}}{{\ell}^{2}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)}{{\ell}^{2}}} \]

      16. unpow2 N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)}{{\ell}^{2}}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)}{{\ell}^{2}}} \]

      18. unpow2 N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\color{blue}{\ell \cdot \ell}}} \]

      19. lower-*.f64 65.2

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\color{blue}{\ell \cdot \ell}}} \]

   8. Simplified 65.2%

      \[\leadsto \frac{2}{t \cdot \color{blue}{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)}{\ell \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}\right)}{\ell \cdot \ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \color{blue}{\left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}}{\ell \cdot \ell}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{\color{blue}{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}}{\ell \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\color{blue}{\ell \cdot \ell}}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell \cdot \ell}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell \cdot \ell} \cdot t}} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell \cdot \ell}} \cdot t} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\color{blue}{\ell \cdot \ell}} \cdot t} \]

      10. associate-/r* N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell}}{\ell}} \cdot t} \]

      11. associate-*l/ N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell} \cdot t}{\ell}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell} \cdot t}{\ell}}} \]

   10. Applied egg-rr 87.6%

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(k \cdot \left(\left(k \cdot t\right) \cdot t\right)\right) \cdot \frac{2}{\ell}\right) \cdot t}{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.9 \cdot 10^{+27}:\\ \;\;\;\;\frac{2}{t \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \frac{k \cdot k}{\ell}\right)}\\ \mathbf{elif}\;t \leq 9.4 \cdot 10^{+119}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot \left(\tan k \cdot \frac{\sin k}{\ell \cdot \ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot \frac{2}{\ell}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+46}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(k, k, 2 \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \left(k \cdot \left(k \cdot \mathsf{fma}\left(\frac{t\_m \cdot \left(k \cdot k\right)}{\ell}, 0.16666666666666666, \frac{t\_m}{\ell}\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(\left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right) \cdot \frac{2}{\ell}\right)}{\ell}}\\ \end{array} \end{array} \]

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 (<= (* l l) 5e+46)
    (/
     2.0
     (/
      (*
       (fma k k (* 2.0 (* t_m t_m)))
       (* k (* k (fma (/ (* t_m (* k k)) l) 0.16666666666666666 (/ t_m l)))))
      l))
    (/ 2.0 (/ (* t_m (* (* k (* t_m (* t_m k))) (/ 2.0 l))) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 5e+46) {
		tmp = 2.0 / ((fma(k, k, (2.0 * (t_m * t_m))) * (k * (k * fma(((t_m * (k * k)) / l), 0.16666666666666666, (t_m / l))))) / l);
	} else {
		tmp = 2.0 / ((t_m * ((k * (t_m * (t_m * k))) * (2.0 / l))) / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 5e+46)
		tmp = Float64(2.0 / Float64(Float64(fma(k, k, Float64(2.0 * Float64(t_m * t_m))) * Float64(k * Float64(k * fma(Float64(Float64(t_m * Float64(k * k)) / l), 0.16666666666666666, Float64(t_m / l))))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(k * Float64(t_m * Float64(t_m * k))) * Float64(2.0 / 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[N[(l * l), $MachinePrecision], 5e+46], N[(2.0 / N[(N[(N[(k * k + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k * N[(N[(N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * 0.16666666666666666 + N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[(N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.0000000000000002e46

   1. Initial program 59.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{t \cdot \left(2 \cdot \color{blue}{\left({t}^{2} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} \]

      4. associate-/l* N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(2 \cdot {t}^{2}\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \color{blue}{{k}^{2} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\right)} \]

      5. distribute-rgt-out N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \left(2 \cdot {t}^{2} + {k}^{2}\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \left(2 \cdot {t}^{2} + {k}^{2}\right)\right)}} \]

   5. Simplified 76.0%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{{\color{blue}{\sin k}}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \left(2 \cdot \left(t \cdot t\right) + k \cdot k\right)\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \left(2 \cdot \left(t \cdot t\right) + k \cdot k\right)\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \color{blue}{\cos k}\right)} \cdot \left(2 \cdot \left(t \cdot t\right) + k \cdot k\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}} \cdot \left(2 \cdot \left(t \cdot t\right) + k \cdot k\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{{\sin k}^{2}}{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}} \cdot \left(2 \cdot \left(t \cdot t\right) + k \cdot k\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \left(2 \cdot \color{blue}{\left(t \cdot t\right)} + k \cdot k\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \left(2 \cdot \left(t \cdot t\right) + \color{blue}{k \cdot k}\right)\right)} \]

      8. lift-fma.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, k \cdot k\right)}\right)} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\frac{{\sin k}^{2} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{t \cdot \frac{{\sin k}^{2} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)}{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{t \cdot \frac{{\sin k}^{2} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)}{\color{blue}{\left(\ell \cdot \cos k\right) \cdot \ell}}} \]

      12. times-frac N/A

          \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{\ell \cdot \cos k} \cdot \frac{\mathsf{fma}\left(2, t \cdot t, k \cdot k\right)}{\ell}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{\ell \cdot \cos k} \cdot \frac{\mathsf{fma}\left(2, t \cdot t, k \cdot k\right)}{\ell}\right)}} \]

   7. Applied egg-rr 94.0%

      \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \frac{\mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}\right)}} \]
   8. Step-by-step derivation

      1. lift-tan.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(\color{blue}{\tan k} \cdot \frac{\sin k}{\ell}\right) \cdot \frac{k \cdot k + 2 \cdot \left(t \cdot t\right)}{\ell}\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(\tan k \cdot \frac{\color{blue}{\sin k}}{\ell}\right) \cdot \frac{k \cdot k + 2 \cdot \left(t \cdot t\right)}{\ell}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(\tan k \cdot \color{blue}{\frac{\sin k}{\ell}}\right) \cdot \frac{k \cdot k + 2 \cdot \left(t \cdot t\right)}{\ell}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\left(\tan k \cdot \frac{\sin k}{\ell}\right)} \cdot \frac{k \cdot k + 2 \cdot \left(t \cdot t\right)}{\ell}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \frac{k \cdot k + 2 \cdot \color{blue}{\left(t \cdot t\right)}}{\ell}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \frac{k \cdot k + \color{blue}{2 \cdot \left(t \cdot t\right)}}{\ell}\right)} \]

      7. lift-fma.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}}{\ell}\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\tan k \cdot \frac{\sin k}{\ell}\right)\right) \cdot \frac{\mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}}} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \frac{\sin k}{\ell}\right)\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}}} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\left(t \cdot \left(\tan k \cdot \frac{\sin k}{\ell}\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\left(t \cdot \left(\tan k \cdot \frac{\sin k}{\ell}\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}}} \]

   9. Applied egg-rr 96.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(\tan k \cdot t\right) \cdot \frac{\sin k}{\ell}\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell} + \frac{t}{\ell}\right)\right)} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]
   11. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell} + \frac{t}{\ell}\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       2. associate-*l* N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot \left(k \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell} + \frac{t}{\ell}\right)\right)\right)} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot \left(k \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell} + \frac{t}{\ell}\right)\right)\right)} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       4. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left(\color{blue}{\frac{{k}^{2} \cdot t}{\ell} \cdot \frac{1}{6}} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       5. associate-/l* N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left(\color{blue}{\left({k}^{2} \cdot \frac{t}{\ell}\right)} \cdot \frac{1}{6} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       6. associate-*r* N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left(\color{blue}{{k}^{2} \cdot \left(\frac{t}{\ell} \cdot \frac{1}{6}\right)} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       7. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left({k}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{t}{\ell}\right)} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot \color{blue}{\left(k \cdot \left({k}^{2} \cdot \left(\frac{1}{6} \cdot \frac{t}{\ell}\right) + \frac{t}{\ell}\right)\right)}\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       9. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left({k}^{2} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{1}{6}\right)} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       10. associate-*r* N/A

           \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left(\color{blue}{\left({k}^{2} \cdot \frac{t}{\ell}\right) \cdot \frac{1}{6}} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       11. associate-/l* N/A

           \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \left(\color{blue}{\frac{{k}^{2} \cdot t}{\ell}} \cdot \frac{1}{6} + \frac{t}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

       12. lower-fma.f64 N/A

           \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot \color{blue}{\mathsf{fma}\left(\frac{{k}^{2} \cdot t}{\ell}, \frac{1}{6}, \frac{t}{\ell}\right)}\right)\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

   12. Simplified 87.0%

       \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot \left(k \cdot \mathsf{fma}\left(\frac{\left(k \cdot k\right) \cdot t}{\ell}, 0.16666666666666666, \frac{t}{\ell}\right)\right)\right)} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}} \]

   if 5.0000000000000002e46 < (*.f64 l l)

   1. Initial program 47.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{t \cdot \left(2 \cdot \color{blue}{\left({t}^{2} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} \]

      4. associate-/l* N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(2 \cdot {t}^{2}\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \color{blue}{{k}^{2} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\right)} \]

      5. distribute-rgt-out N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \left(2 \cdot {t}^{2} + {k}^{2}\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \left(2 \cdot {t}^{2} + {k}^{2}\right)\right)}} \]

   5. Simplified 64.8%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{t \cdot \color{blue}{\left(2 \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\frac{2 \cdot \left({k}^{2} \cdot {t}^{2}\right)}{{\ell}^{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{t \cdot \frac{\color{blue}{\left({k}^{2} \cdot {t}^{2}\right) \cdot 2}}{{\ell}^{2}}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{t \cdot \frac{\color{blue}{{k}^{2} \cdot \left({t}^{2} \cdot 2\right)}}{{\ell}^{2}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{2} \cdot \color{blue}{\left(2 \cdot {t}^{2}\right)}}{{\ell}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\frac{{k}^{2} \cdot \left(2 \cdot {t}^{2}\right)}{{\ell}^{2}}}} \]

      6. *-commutative N/A

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{2} \cdot \color{blue}{\left({t}^{2} \cdot 2\right)}}{{\ell}^{2}}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{t \cdot \frac{\color{blue}{\left({k}^{2} \cdot {t}^{2}\right) \cdot 2}}{{\ell}^{2}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{t \cdot \frac{\color{blue}{2 \cdot \left({k}^{2} \cdot {t}^{2}\right)}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{\color{blue}{2 \cdot \left({k}^{2} \cdot {t}^{2}\right)}}{{\ell}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \color{blue}{\left({t}^{2} \cdot {k}^{2}\right)}}{{\ell}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot {k}^{2}\right)}{{\ell}^{2}}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \color{blue}{\left(t \cdot \left(t \cdot {k}^{2}\right)\right)}}{{\ell}^{2}}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)}{{\ell}^{2}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \color{blue}{\left(t \cdot \left({k}^{2} \cdot t\right)\right)}}{{\ell}^{2}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)}{{\ell}^{2}}} \]

      16. unpow2 N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)}{{\ell}^{2}}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)}{{\ell}^{2}}} \]

      18. unpow2 N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\color{blue}{\ell \cdot \ell}}} \]

      19. lower-*.f64 58.5

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\color{blue}{\ell \cdot \ell}}} \]

   8. Simplified 58.5%

      \[\leadsto \frac{2}{t \cdot \color{blue}{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)}{\ell \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}\right)}{\ell \cdot \ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \color{blue}{\left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}}{\ell \cdot \ell}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{\color{blue}{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}}{\ell \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\color{blue}{\ell \cdot \ell}}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell \cdot \ell}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell \cdot \ell} \cdot t}} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell \cdot \ell}} \cdot t} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\color{blue}{\ell \cdot \ell}} \cdot t} \]

      10. associate-/r* N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell}}{\ell}} \cdot t} \]

      11. associate-*l/ N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell} \cdot t}{\ell}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell} \cdot t}{\ell}}} \]

   10. Applied egg-rr 68.3%

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(k \cdot \left(\left(k \cdot t\right) \cdot t\right)\right) \cdot \frac{2}{\ell}\right) \cdot t}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+46}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot \left(k \cdot \mathsf{fma}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell}, 0.16666666666666666, \frac{t}{\ell}\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot \frac{2}{\ell}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.0% accurate, 6.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.05 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left(\mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}\right)}\\ \mathbf{elif}\;t\_m \leq 5 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{t\_m \cdot t\_m} \cdot \frac{\ell}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(\left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right) \cdot \frac{2}{\ell}\right)}{\ell}}\\ \end{array} \end{array} \]

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 (<= t_m 1.05e-113)
    (/ 2.0 (* t_m (* (fma 2.0 (* t_m t_m) (* k k)) (/ (* k k) (* l l)))))
    (if (<= t_m 5e+103)
      (* (/ (/ l k) (* t_m t_m)) (/ l (* t_m k)))
      (/ 2.0 (/ (* t_m (* (* k (* t_m (* t_m k))) (/ 2.0 l))) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.05e-113) {
		tmp = 2.0 / (t_m * (fma(2.0, (t_m * t_m), (k * k)) * ((k * k) / (l * l))));
	} else if (t_m <= 5e+103) {
		tmp = ((l / k) / (t_m * t_m)) * (l / (t_m * k));
	} else {
		tmp = 2.0 / ((t_m * ((k * (t_m * (t_m * k))) * (2.0 / l))) / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.05e-113)
		tmp = Float64(2.0 / Float64(t_m * Float64(fma(2.0, Float64(t_m * t_m), Float64(k * k)) * Float64(Float64(k * k) / Float64(l * l)))));
	elseif (t_m <= 5e+103)
		tmp = Float64(Float64(Float64(l / k) / Float64(t_m * t_m)) * Float64(l / Float64(t_m * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(k * Float64(t_m * Float64(t_m * k))) * Float64(2.0 / 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[t$95$m, 1.05e-113], N[(2.0 / N[(t$95$m * N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5e+103], N[(N[(N[(l / k), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[(N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.05e-113

   1. Initial program 48.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{t \cdot \left(2 \cdot \color{blue}{\left({t}^{2} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} \]

      4. associate-/l* N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(2 \cdot {t}^{2}\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \color{blue}{{k}^{2} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\right)} \]

      5. distribute-rgt-out N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \left(2 \cdot {t}^{2} + {k}^{2}\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \left(2 \cdot {t}^{2} + {k}^{2}\right)\right)}} \]

   5. Simplified 71.5%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\frac{{k}^{2}}{{\ell}^{2}}} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\frac{{k}^{2}}{{\ell}^{2}}} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{\color{blue}{k \cdot k}}{{\ell}^{2}} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{\color{blue}{k \cdot k}}{{\ell}^{2}} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{k \cdot k}{\color{blue}{\ell \cdot \ell}} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]

      5. lower-*.f64 64.8

         \[\leadsto \frac{2}{t \cdot \left(\frac{k \cdot k}{\color{blue}{\ell \cdot \ell}} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]

   8. Simplified 64.8%

      \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\frac{k \cdot k}{\ell \cdot \ell}} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]

   if 1.05e-113 < t < 5e103

   1. Initial program 67.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   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. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{2 \cdot {t}^{3}}{{\ell}^{2}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{2 \cdot {t}^{3}}{{\ell}^{2}}}} \]

      10. unpow3 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{2 \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}}{{\ell}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{2 \cdot \left(\color{blue}{{t}^{2}} \cdot t\right)}{{\ell}^{2}}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right)} \cdot t}{{\ell}^{2}}} \]

      15. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t}{{\ell}^{2}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t}{{\ell}^{2}}} \]

      17. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

      18. lower-*.f64 51.7

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

   5. Simplified 51.7%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\ell \cdot \ell}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      8. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {t}^{2}\right)}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

      12. lower-*.f64 54.2

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   8. Simplified 54.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{\color{blue}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(k \cdot t\right)}} \]

      10. times-frac N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{t \cdot t} \cdot \frac{\ell}{k \cdot t}} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{t \cdot t} \cdot \frac{\ell}{k \cdot t}} \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{t \cdot t}} \cdot \frac{\ell}{k \cdot t} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\ell}{k}}}{t \cdot t} \cdot \frac{\ell}{k \cdot t} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\ell}{k}}{t \cdot t} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

      15. lower-*.f64 64.0

          \[\leadsto \frac{\frac{\ell}{k}}{t \cdot t} \cdot \frac{\ell}{\color{blue}{k \cdot t}} \]

   10. Applied egg-rr 64.0%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{t \cdot t} \cdot \frac{\ell}{k \cdot t}} \]

   if 5e103 < t

   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 t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{t \cdot \left(2 \cdot \color{blue}{\left({t}^{2} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} \]

      4. associate-/l* N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(2 \cdot {t}^{2}\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \color{blue}{{k}^{2} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\right)} \]

      5. distribute-rgt-out N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \left(2 \cdot {t}^{2} + {k}^{2}\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \left(2 \cdot {t}^{2} + {k}^{2}\right)\right)}} \]

   5. Simplified 64.2%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{t \cdot \color{blue}{\left(2 \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\frac{2 \cdot \left({k}^{2} \cdot {t}^{2}\right)}{{\ell}^{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{t \cdot \frac{\color{blue}{\left({k}^{2} \cdot {t}^{2}\right) \cdot 2}}{{\ell}^{2}}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{t \cdot \frac{\color{blue}{{k}^{2} \cdot \left({t}^{2} \cdot 2\right)}}{{\ell}^{2}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{2} \cdot \color{blue}{\left(2 \cdot {t}^{2}\right)}}{{\ell}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\frac{{k}^{2} \cdot \left(2 \cdot {t}^{2}\right)}{{\ell}^{2}}}} \]

      6. *-commutative N/A

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{2} \cdot \color{blue}{\left({t}^{2} \cdot 2\right)}}{{\ell}^{2}}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{t \cdot \frac{\color{blue}{\left({k}^{2} \cdot {t}^{2}\right) \cdot 2}}{{\ell}^{2}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{t \cdot \frac{\color{blue}{2 \cdot \left({k}^{2} \cdot {t}^{2}\right)}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{\color{blue}{2 \cdot \left({k}^{2} \cdot {t}^{2}\right)}}{{\ell}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \color{blue}{\left({t}^{2} \cdot {k}^{2}\right)}}{{\ell}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot {k}^{2}\right)}{{\ell}^{2}}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \color{blue}{\left(t \cdot \left(t \cdot {k}^{2}\right)\right)}}{{\ell}^{2}}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)}{{\ell}^{2}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \color{blue}{\left(t \cdot \left({k}^{2} \cdot t\right)\right)}}{{\ell}^{2}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)}{{\ell}^{2}}} \]

      16. unpow2 N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)}{{\ell}^{2}}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)}{{\ell}^{2}}} \]

      18. unpow2 N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\color{blue}{\ell \cdot \ell}}} \]

      19. lower-*.f64 67.0

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\color{blue}{\ell \cdot \ell}}} \]

   8. Simplified 67.0%

      \[\leadsto \frac{2}{t \cdot \color{blue}{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)}{\ell \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}\right)}{\ell \cdot \ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \color{blue}{\left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}}{\ell \cdot \ell}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{\color{blue}{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}}{\ell \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\color{blue}{\ell \cdot \ell}}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell \cdot \ell}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell \cdot \ell} \cdot t}} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell \cdot \ell}} \cdot t} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\color{blue}{\ell \cdot \ell}} \cdot t} \]

      10. associate-/r* N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell}}{\ell}} \cdot t} \]

      11. associate-*l/ N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell} \cdot t}{\ell}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell} \cdot t}{\ell}}} \]

   10. Applied egg-rr 86.8%

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(k \cdot \left(\left(k \cdot t\right) \cdot t\right)\right) \cdot \frac{2}{\ell}\right) \cdot t}{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{t \cdot \left(\mathsf{fma}\left(2, t \cdot t, k \cdot k\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{t \cdot t} \cdot \frac{\ell}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot \frac{2}{\ell}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.1% accurate, 6.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.5 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left(\frac{\mathsf{fma}\left(k, k, 2 \cdot \left(t\_m \cdot t\_m\right)\right)}{\ell} \cdot \frac{k \cdot k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(\left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right) \cdot \frac{2}{\ell}\right)}{\ell}}\\ \end{array} \end{array} \]

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 (<= t_m 8.5e-22)
    (/ 2.0 (* t_m (* (/ (fma k k (* 2.0 (* t_m t_m))) l) (/ (* k k) l))))
    (/ 2.0 (/ (* t_m (* (* k (* t_m (* t_m k))) (/ 2.0 l))) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8.5e-22) {
		tmp = 2.0 / (t_m * ((fma(k, k, (2.0 * (t_m * t_m))) / l) * ((k * k) / l)));
	} else {
		tmp = 2.0 / ((t_m * ((k * (t_m * (t_m * k))) * (2.0 / l))) / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8.5e-22)
		tmp = Float64(2.0 / Float64(t_m * Float64(Float64(fma(k, k, Float64(2.0 * Float64(t_m * t_m))) / l) * Float64(Float64(k * k) / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(k * Float64(t_m * Float64(t_m * k))) * Float64(2.0 / 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[t$95$m, 8.5e-22], N[(2.0 / N[(t$95$m * N[(N[(N[(k * k + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[(N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.5000000000000001e-22

   1. Initial program 49.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{t \cdot \left(2 \cdot \color{blue}{\left({t}^{2} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} \]

      4. associate-/l* N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(2 \cdot {t}^{2}\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \color{blue}{{k}^{2} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\right)} \]

      5. distribute-rgt-out N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \left(2 \cdot {t}^{2} + {k}^{2}\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \left(2 \cdot {t}^{2} + {k}^{2}\right)\right)}} \]

   5. Simplified 70.7%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{{\color{blue}{\sin k}}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \left(2 \cdot \left(t \cdot t\right) + k \cdot k\right)\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \left(2 \cdot \left(t \cdot t\right) + k \cdot k\right)\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \color{blue}{\cos k}\right)} \cdot \left(2 \cdot \left(t \cdot t\right) + k \cdot k\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}} \cdot \left(2 \cdot \left(t \cdot t\right) + k \cdot k\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{{\sin k}^{2}}{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}} \cdot \left(2 \cdot \left(t \cdot t\right) + k \cdot k\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \left(2 \cdot \color{blue}{\left(t \cdot t\right)} + k \cdot k\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \left(2 \cdot \left(t \cdot t\right) + \color{blue}{k \cdot k}\right)\right)} \]

      8. lift-fma.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, k \cdot k\right)}\right)} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\frac{{\sin k}^{2} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{t \cdot \frac{{\sin k}^{2} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)}{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{t \cdot \frac{{\sin k}^{2} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)}{\color{blue}{\left(\ell \cdot \cos k\right) \cdot \ell}}} \]

      12. times-frac N/A

          \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{\ell \cdot \cos k} \cdot \frac{\mathsf{fma}\left(2, t \cdot t, k \cdot k\right)}{\ell}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{\ell \cdot \cos k} \cdot \frac{\mathsf{fma}\left(2, t \cdot t, k \cdot k\right)}{\ell}\right)}} \]

   7. Applied egg-rr 82.3%

      \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \frac{\mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}\right)}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\frac{{k}^{2}}{\ell}} \cdot \frac{\mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}\right)} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\frac{{k}^{2}}{\ell}} \cdot \frac{\mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{\mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}\right)} \]

      3. lower-*.f64 69.6

         \[\leadsto \frac{2}{t \cdot \left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{\mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}\right)} \]

   10. Simplified 69.6%

       \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\frac{k \cdot k}{\ell}} \cdot \frac{\mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell}\right)} \]

   if 8.5000000000000001e-22 < t

   1. Initial program 65.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{t \cdot \left(2 \cdot \color{blue}{\left({t}^{2} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} \]

      4. associate-/l* N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(2 \cdot {t}^{2}\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \color{blue}{{k}^{2} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\right)} \]

      5. distribute-rgt-out N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \left(2 \cdot {t}^{2} + {k}^{2}\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \left(2 \cdot {t}^{2} + {k}^{2}\right)\right)}} \]

   5. Simplified 69.5%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{t \cdot \color{blue}{\left(2 \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\frac{2 \cdot \left({k}^{2} \cdot {t}^{2}\right)}{{\ell}^{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{t \cdot \frac{\color{blue}{\left({k}^{2} \cdot {t}^{2}\right) \cdot 2}}{{\ell}^{2}}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{t \cdot \frac{\color{blue}{{k}^{2} \cdot \left({t}^{2} \cdot 2\right)}}{{\ell}^{2}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{2} \cdot \color{blue}{\left(2 \cdot {t}^{2}\right)}}{{\ell}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\frac{{k}^{2} \cdot \left(2 \cdot {t}^{2}\right)}{{\ell}^{2}}}} \]

      6. *-commutative N/A

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{2} \cdot \color{blue}{\left({t}^{2} \cdot 2\right)}}{{\ell}^{2}}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{t \cdot \frac{\color{blue}{\left({k}^{2} \cdot {t}^{2}\right) \cdot 2}}{{\ell}^{2}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{t \cdot \frac{\color{blue}{2 \cdot \left({k}^{2} \cdot {t}^{2}\right)}}{{\ell}^{2}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{\color{blue}{2 \cdot \left({k}^{2} \cdot {t}^{2}\right)}}{{\ell}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \color{blue}{\left({t}^{2} \cdot {k}^{2}\right)}}{{\ell}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot {k}^{2}\right)}{{\ell}^{2}}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \color{blue}{\left(t \cdot \left(t \cdot {k}^{2}\right)\right)}}{{\ell}^{2}}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)}{{\ell}^{2}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \color{blue}{\left(t \cdot \left({k}^{2} \cdot t\right)\right)}}{{\ell}^{2}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)}{{\ell}^{2}}} \]

      16. unpow2 N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)}{{\ell}^{2}}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)}{{\ell}^{2}}} \]

      18. unpow2 N/A

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\color{blue}{\ell \cdot \ell}}} \]

      19. lower-*.f64 64.7

          \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\color{blue}{\ell \cdot \ell}}} \]

   8. Simplified 64.7%

      \[\leadsto \frac{2}{t \cdot \color{blue}{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell \cdot \ell}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)}{\ell \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}\right)}{\ell \cdot \ell}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \color{blue}{\left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}}{\ell \cdot \ell}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{\color{blue}{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}}{\ell \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\color{blue}{\ell \cdot \ell}}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell \cdot \ell}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell \cdot \ell} \cdot t}} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell \cdot \ell}} \cdot t} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\color{blue}{\ell \cdot \ell}} \cdot t} \]

      10. associate-/r* N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell}}{\ell}} \cdot t} \]

      11. associate-*l/ N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell} \cdot t}{\ell}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{2 \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}{\ell} \cdot t}{\ell}}} \]

   10. Applied egg-rr 77.2%

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(k \cdot \left(\left(k \cdot t\right) \cdot t\right)\right) \cdot \frac{2}{\ell}\right) \cdot t}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{t \cdot \left(\frac{\mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)}{\ell} \cdot \frac{k \cdot k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot \frac{2}{\ell}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.4% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.05 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left(\mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}\right)}\\ \mathbf{elif}\;t\_m \leq 2.5 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{t\_m \cdot t\_m} \cdot \frac{\ell}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.05e-113)
    (/ 2.0 (* t_m (* (fma 2.0 (* t_m t_m) (* k k)) (/ (* k k) (* l l)))))
    (if (<= t_m 2.5e+123)
      (* (/ (/ l k) (* t_m t_m)) (/ l (* t_m k)))
      (* l (/ l (* t_m (* k (* t_m (* t_m k))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.05e-113) {
		tmp = 2.0 / (t_m * (fma(2.0, (t_m * t_m), (k * k)) * ((k * k) / (l * l))));
	} else if (t_m <= 2.5e+123) {
		tmp = ((l / k) / (t_m * t_m)) * (l / (t_m * k));
	} else {
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.05e-113)
		tmp = Float64(2.0 / Float64(t_m * Float64(fma(2.0, Float64(t_m * t_m), Float64(k * k)) * Float64(Float64(k * k) / Float64(l * l)))));
	elseif (t_m <= 2.5e+123)
		tmp = Float64(Float64(Float64(l / k) / Float64(t_m * t_m)) * Float64(l / Float64(t_m * k)));
	else
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(k * Float64(t_m * Float64(t_m * k))))));
	end
	return Float64(t_s * tmp)
end

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, 1.05e-113], N[(2.0 / N[(t$95$m * N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.5e+123], N[(N[(N[(l / k), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(t$95$m * N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.05e-113

   1. Initial program 48.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{t \cdot \left(2 \cdot \color{blue}{\left({t}^{2} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)} \]

      4. associate-/l* N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(2 \cdot {t}^{2}\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \color{blue}{{k}^{2} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\right)} \]

      5. distribute-rgt-out N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \left(2 \cdot {t}^{2} + {k}^{2}\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot \left(2 \cdot {t}^{2} + {k}^{2}\right)\right)}} \]

   5. Simplified 71.5%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\frac{{k}^{2}}{{\ell}^{2}}} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\frac{{k}^{2}}{{\ell}^{2}}} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{\color{blue}{k \cdot k}}{{\ell}^{2}} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{\color{blue}{k \cdot k}}{{\ell}^{2}} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2}{t \cdot \left(\frac{k \cdot k}{\color{blue}{\ell \cdot \ell}} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]

      5. lower-*.f64 64.8

         \[\leadsto \frac{2}{t \cdot \left(\frac{k \cdot k}{\color{blue}{\ell \cdot \ell}} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]

   8. Simplified 64.8%

      \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\frac{k \cdot k}{\ell \cdot \ell}} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]

   if 1.05e-113 < t < 2.49999999999999987e123

   1. Initial program 65.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   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. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{2 \cdot {t}^{3}}{{\ell}^{2}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{2 \cdot {t}^{3}}{{\ell}^{2}}}} \]

      10. unpow3 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{2 \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}}{{\ell}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{2 \cdot \left(\color{blue}{{t}^{2}} \cdot t\right)}{{\ell}^{2}}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right)} \cdot t}{{\ell}^{2}}} \]

      15. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t}{{\ell}^{2}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t}{{\ell}^{2}}} \]

      17. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

      18. lower-*.f64 51.7

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

   5. Simplified 51.7%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\ell \cdot \ell}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      8. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {t}^{2}\right)}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

      12. lower-*.f64 53.9

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   8. Simplified 53.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{\color{blue}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(k \cdot t\right)}} \]

      10. times-frac N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{t \cdot t} \cdot \frac{\ell}{k \cdot t}} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{t \cdot t} \cdot \frac{\ell}{k \cdot t}} \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{t \cdot t}} \cdot \frac{\ell}{k \cdot t} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\ell}{k}}}{t \cdot t} \cdot \frac{\ell}{k \cdot t} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\ell}{k}}{t \cdot t} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

      15. lower-*.f64 66.5

          \[\leadsto \frac{\frac{\ell}{k}}{t \cdot t} \cdot \frac{\ell}{\color{blue}{k \cdot t}} \]

   10. Applied egg-rr 66.5%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{t \cdot t} \cdot \frac{\ell}{k \cdot t}} \]

   if 2.49999999999999987e123 < t

   1. Initial program 61.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{2 \cdot {t}^{3}}{{\ell}^{2}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{2 \cdot {t}^{3}}{{\ell}^{2}}}} \]

      10. unpow3 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{2 \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}}{{\ell}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{2 \cdot \left(\color{blue}{{t}^{2}} \cdot t\right)}{{\ell}^{2}}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right)} \cdot t}{{\ell}^{2}}} \]

      15. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t}{{\ell}^{2}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t}{{\ell}^{2}}} \]

      17. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

      18. lower-*.f64 58.4

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

   5. Simplified 58.4%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\ell \cdot \ell}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      8. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {t}^{2}\right)}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

      12. lower-*.f64 61.3

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   8. Simplified 61.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

      8. lower-/.f64 66.6

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \cdot \ell \]

   10. Applied egg-rr 66.6%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \cdot \ell \]

       3. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

       6. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \cdot \ell \]

       7. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t}} \cdot \ell \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t} \cdot \ell \]

       9. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right)} \cdot t} \cdot \ell \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\left(\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot t\right) \cdot t} \cdot \ell \]

       11. *-commutative N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot t} \cdot \ell \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot t} \cdot \ell \]

       13. lower-*.f64 71.6

           \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right) \cdot t}} \cdot \ell \]

       14. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot t} \cdot \ell \]

       15. *-commutative N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right)} \cdot t} \cdot \ell \]

       16. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\left(\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot t\right) \cdot t} \cdot \ell \]

       17. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\left(\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot t\right) \cdot t} \cdot \ell \]

       18. associate-*l* N/A

           \[\leadsto \frac{\ell}{\left(\color{blue}{\left(k \cdot \left(k \cdot t\right)\right)} \cdot t\right) \cdot t} \cdot \ell \]

       19. associate-*l* N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(k \cdot t\right) \cdot t\right)\right)} \cdot t} \cdot \ell \]

       20. lower-*.f64 N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(k \cdot t\right) \cdot t\right)\right)} \cdot t} \cdot \ell \]

       21. lower-*.f64 N/A

           \[\leadsto \frac{\ell}{\left(k \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right) \cdot t} \cdot \ell \]

       22. lower-*.f64 81.8

           \[\leadsto \frac{\ell}{\left(k \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot t\right)\right) \cdot t} \cdot \ell \]

   12. Applied egg-rr 81.8%

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(k \cdot t\right) \cdot t\right)\right) \cdot t}} \cdot \ell \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{t \cdot \left(\mathsf{fma}\left(2, t \cdot t, k \cdot k\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}\right)}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{t \cdot t} \cdot \frac{\ell}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 68.6% accurate, 7.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \left(t\_m \cdot k\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+102}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{t\_2} \cdot \frac{\ell}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(k \cdot t\_2\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* t_m (* t_m k))))
   (*
    t_s
    (if (<= (* l l) 1e+102)
      (* (/ (/ l k) t_2) (/ l t_m))
      (* l (/ l (* t_m (* k t_2))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * (t_m * k);
	double tmp;
	if ((l * l) <= 1e+102) {
		tmp = ((l / k) / t_2) * (l / t_m);
	} else {
		tmp = l * (l / (t_m * (k * t_2)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * (t_m * k)
    if ((l * l) <= 1d+102) then
        tmp = ((l / k) / t_2) * (l / t_m)
    else
        tmp = l * (l / (t_m * (k * t_2)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * (t_m * k);
	double tmp;
	if ((l * l) <= 1e+102) {
		tmp = ((l / k) / t_2) * (l / t_m);
	} else {
		tmp = l * (l / (t_m * (k * t_2)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = t_m * (t_m * k)
	tmp = 0
	if (l * l) <= 1e+102:
		tmp = ((l / k) / t_2) * (l / t_m)
	else:
		tmp = l * (l / (t_m * (k * t_2)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m * Float64(t_m * k))
	tmp = 0.0
	if (Float64(l * l) <= 1e+102)
		tmp = Float64(Float64(Float64(l / k) / t_2) * Float64(l / t_m));
	else
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(k * t_2))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = t_m * (t_m * k);
	tmp = 0.0;
	if ((l * l) <= 1e+102)
		tmp = ((l / k) / t_2) * (l / t_m);
	else
		tmp = l * (l / (t_m * (k * t_2)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e+102], N[(N[(N[(l / k), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(t$95$m * N[(k * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 9.99999999999999977e101

   1. Initial program 60.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   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. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{2 \cdot {t}^{3}}{{\ell}^{2}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{2 \cdot {t}^{3}}{{\ell}^{2}}}} \]

      10. unpow3 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{2 \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}}{{\ell}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{2 \cdot \left(\color{blue}{{t}^{2}} \cdot t\right)}{{\ell}^{2}}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right)} \cdot t}{{\ell}^{2}}} \]

      15. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t}{{\ell}^{2}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t}{{\ell}^{2}}} \]

      17. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

      18. lower-*.f64 53.6

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

   5. Simplified 53.6%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\ell \cdot \ell}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      8. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {t}^{2}\right)}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

      12. lower-*.f64 60.9

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   8. Simplified 60.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{\color{blue}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      10. associate-*r* N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot t\right) \cdot t}} \]

      11. times-frac N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{\left(k \cdot t\right) \cdot t} \cdot \frac{\ell}{t}} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{\left(k \cdot t\right) \cdot t} \cdot \frac{\ell}{t}} \]

      13. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{\left(k \cdot t\right) \cdot t}} \cdot \frac{\ell}{t} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\ell}{k}}}{\left(k \cdot t\right) \cdot t} \cdot \frac{\ell}{t} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{k}}{\color{blue}{\left(k \cdot t\right) \cdot t}} \cdot \frac{\ell}{t} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{k}}{\color{blue}{\left(k \cdot t\right)} \cdot t} \cdot \frac{\ell}{t} \]

      17. lower-/.f64 76.8

          \[\leadsto \frac{\frac{\ell}{k}}{\left(k \cdot t\right) \cdot t} \cdot \color{blue}{\frac{\ell}{t}} \]

   10. Applied egg-rr 76.8%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{\left(k \cdot t\right) \cdot t} \cdot \frac{\ell}{t}} \]

   if 9.99999999999999977e101 < (*.f64 l l)

   1. Initial program 46.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{2 \cdot {t}^{3}}{{\ell}^{2}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{2 \cdot {t}^{3}}{{\ell}^{2}}}} \]

      10. unpow3 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{2 \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}}{{\ell}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{2 \cdot \left(\color{blue}{{t}^{2}} \cdot t\right)}{{\ell}^{2}}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right)} \cdot t}{{\ell}^{2}}} \]

      15. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t}{{\ell}^{2}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t}{{\ell}^{2}}} \]

      17. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

      18. lower-*.f64 51.5

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

   5. Simplified 51.5%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\ell \cdot \ell}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      8. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {t}^{2}\right)}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

      12. lower-*.f64 55.1

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   8. Simplified 55.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

      8. lower-/.f64 57.4

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \cdot \ell \]

   10. Applied egg-rr 57.4%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \cdot \ell \]

       3. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

       6. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \cdot \ell \]

       7. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t}} \cdot \ell \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t} \cdot \ell \]

       9. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right)} \cdot t} \cdot \ell \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\left(\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot t\right) \cdot t} \cdot \ell \]

       11. *-commutative N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot t} \cdot \ell \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot t} \cdot \ell \]

       13. lower-*.f64 60.0

           \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right) \cdot t}} \cdot \ell \]

       14. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot t} \cdot \ell \]

       15. *-commutative N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right)} \cdot t} \cdot \ell \]

       16. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\left(\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot t\right) \cdot t} \cdot \ell \]

       17. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\left(\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot t\right) \cdot t} \cdot \ell \]

       18. associate-*l* N/A

           \[\leadsto \frac{\ell}{\left(\color{blue}{\left(k \cdot \left(k \cdot t\right)\right)} \cdot t\right) \cdot t} \cdot \ell \]

       19. associate-*l* N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(k \cdot t\right) \cdot t\right)\right)} \cdot t} \cdot \ell \]

       20. lower-*.f64 N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(k \cdot t\right) \cdot t\right)\right)} \cdot t} \cdot \ell \]

       21. lower-*.f64 N/A

           \[\leadsto \frac{\ell}{\left(k \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right) \cdot t} \cdot \ell \]

       22. lower-*.f64 66.5

           \[\leadsto \frac{\ell}{\left(k \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot t\right)\right) \cdot t} \cdot \ell \]

   12. Applied egg-rr 66.5%

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(k \cdot t\right) \cdot t\right)\right) \cdot t}} \cdot \ell \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+102}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{t \cdot \left(t \cdot k\right)} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 69.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \left(t\_m \cdot k\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+47}:\\ \;\;\;\;\frac{\ell}{t\_2} \cdot \frac{\frac{\ell}{k}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(k \cdot t\_2\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* t_m (* t_m k))))
   (*
    t_s
    (if (<= (* l l) 5e+47)
      (* (/ l t_2) (/ (/ l k) t_m))
      (* l (/ l (* t_m (* k t_2))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * (t_m * k);
	double tmp;
	if ((l * l) <= 5e+47) {
		tmp = (l / t_2) * ((l / k) / t_m);
	} else {
		tmp = l * (l / (t_m * (k * t_2)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * (t_m * k)
    if ((l * l) <= 5d+47) then
        tmp = (l / t_2) * ((l / k) / t_m)
    else
        tmp = l * (l / (t_m * (k * t_2)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * (t_m * k);
	double tmp;
	if ((l * l) <= 5e+47) {
		tmp = (l / t_2) * ((l / k) / t_m);
	} else {
		tmp = l * (l / (t_m * (k * t_2)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = t_m * (t_m * k)
	tmp = 0
	if (l * l) <= 5e+47:
		tmp = (l / t_2) * ((l / k) / t_m)
	else:
		tmp = l * (l / (t_m * (k * t_2)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m * Float64(t_m * k))
	tmp = 0.0
	if (Float64(l * l) <= 5e+47)
		tmp = Float64(Float64(l / t_2) * Float64(Float64(l / k) / t_m));
	else
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(k * t_2))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = t_m * (t_m * k);
	tmp = 0.0;
	if ((l * l) <= 5e+47)
		tmp = (l / t_2) * ((l / k) / t_m);
	else
		tmp = l * (l / (t_m * (k * t_2)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 5e+47], N[(N[(l / t$95$2), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(t$95$m * N[(k * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.00000000000000022e47

   1. Initial program 59.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   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. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{2 \cdot {t}^{3}}{{\ell}^{2}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{2 \cdot {t}^{3}}{{\ell}^{2}}}} \]

      10. unpow3 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{2 \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}}{{\ell}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{2 \cdot \left(\color{blue}{{t}^{2}} \cdot t\right)}{{\ell}^{2}}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right)} \cdot t}{{\ell}^{2}}} \]

      15. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t}{{\ell}^{2}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t}{{\ell}^{2}}} \]

      17. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

      18. lower-*.f64 53.0

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

   5. Simplified 53.0%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\ell \cdot \ell}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      8. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {t}^{2}\right)}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

      12. lower-*.f64 60.9

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   8. Simplified 60.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k} \cdot \frac{\ell}{\color{blue}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      6. associate-/r* N/A

         \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{k}}{t \cdot \left(t \cdot t\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{k}}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \frac{\ell}{k}}{k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \frac{\ell}{k}}{\color{blue}{\left(k \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \frac{\ell}{k}}{\left(k \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      11. associate-*r* N/A

          \[\leadsto \frac{\ell \cdot \frac{\ell}{k}}{\color{blue}{\left(\left(k \cdot t\right) \cdot t\right) \cdot t}} \]

      12. times-frac N/A

          \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot t\right) \cdot t} \cdot \frac{\frac{\ell}{k}}{t}} \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot t\right) \cdot t} \cdot \frac{\frac{\ell}{k}}{t}} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot t\right) \cdot t}} \cdot \frac{\frac{\ell}{k}}{t} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot t}} \cdot \frac{\frac{\ell}{k}}{t} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot t} \cdot \frac{\frac{\ell}{k}}{t} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot t} \cdot \color{blue}{\frac{\frac{\ell}{k}}{t}} \]

      18. lower-/.f64 77.2

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot t} \cdot \frac{\color{blue}{\frac{\ell}{k}}}{t} \]

   10. Applied egg-rr 77.2%

       \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot t\right) \cdot t} \cdot \frac{\frac{\ell}{k}}{t}} \]

   if 5.00000000000000022e47 < (*.f64 l l)

   1. Initial program 48.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   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. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{2 \cdot {t}^{3}}{{\ell}^{2}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{2 \cdot {t}^{3}}{{\ell}^{2}}}} \]

      10. unpow3 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{2 \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}}{{\ell}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{2 \cdot \left(\color{blue}{{t}^{2}} \cdot t\right)}{{\ell}^{2}}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right)} \cdot t}{{\ell}^{2}}} \]

      15. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t}{{\ell}^{2}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t}{{\ell}^{2}}} \]

      17. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

      18. lower-*.f64 52.2

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

   5. Simplified 52.2%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\ell \cdot \ell}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      8. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {t}^{2}\right)}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

      12. lower-*.f64 55.6

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   8. Simplified 55.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

      8. lower-/.f64 57.7

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \cdot \ell \]

   10. Applied egg-rr 57.7%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \cdot \ell \]

       3. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

       6. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \cdot \ell \]

       7. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t}} \cdot \ell \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t} \cdot \ell \]

       9. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right)} \cdot t} \cdot \ell \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\left(\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot t\right) \cdot t} \cdot \ell \]

       11. *-commutative N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot t} \cdot \ell \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot t} \cdot \ell \]

       13. lower-*.f64 60.8

           \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right) \cdot t}} \cdot \ell \]

       14. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot t} \cdot \ell \]

       15. *-commutative N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right)} \cdot t} \cdot \ell \]

       16. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\left(\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot t\right) \cdot t} \cdot \ell \]

       17. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\left(\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot t\right) \cdot t} \cdot \ell \]

       18. associate-*l* N/A

           \[\leadsto \frac{\ell}{\left(\color{blue}{\left(k \cdot \left(k \cdot t\right)\right)} \cdot t\right) \cdot t} \cdot \ell \]

       19. associate-*l* N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(k \cdot t\right) \cdot t\right)\right)} \cdot t} \cdot \ell \]

       20. lower-*.f64 N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(k \cdot t\right) \cdot t\right)\right)} \cdot t} \cdot \ell \]

       21. lower-*.f64 N/A

           \[\leadsto \frac{\ell}{\left(k \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right) \cdot t} \cdot \ell \]

       22. lower-*.f64 66.8

           \[\leadsto \frac{\ell}{\left(k \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot t\right)\right) \cdot t} \cdot \ell \]

   12. Applied egg-rr 66.8%

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(k \cdot t\right) \cdot t\right)\right) \cdot t}} \cdot \ell \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+47}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot k\right)} \cdot \frac{\frac{\ell}{k}}{t}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 66.7% accurate, 8.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-161}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\ell}{k \cdot \left(t\_m \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 2e-161)
    (* (/ l (* t_m k)) (/ l (* k (* t_m t_m))))
    (* l (/ l (* t_m (* k (* t_m (* t_m k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 2e-161) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else {
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 2d-161) then
        tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)))
    else
        tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 2e-161) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else {
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 2e-161:
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)))
	else:
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 2e-161)
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(l / Float64(k * Float64(t_m * t_m))));
	else
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(k * Float64(t_m * Float64(t_m * k))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 2e-161)
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	else
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 2e-161], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(t$95$m * N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 2.00000000000000006e-161

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   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. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{2 \cdot {t}^{3}}{{\ell}^{2}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{2 \cdot {t}^{3}}{{\ell}^{2}}}} \]

      10. unpow3 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{2 \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}}{{\ell}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{2 \cdot \left(\color{blue}{{t}^{2}} \cdot t\right)}{{\ell}^{2}}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right)} \cdot t}{{\ell}^{2}}} \]

      15. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t}{{\ell}^{2}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t}{{\ell}^{2}}} \]

      17. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

      18. lower-*.f64 48.3

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

   5. Simplified 48.3%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\ell \cdot \ell}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      8. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {t}^{2}\right)}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

      12. lower-*.f64 59.4

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   8. Simplified 59.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right) \cdot k}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot k} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right) \cdot k} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(t \cdot t\right)\right)} \cdot k} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot k\right)}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot t}} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{k \cdot t}} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot k}} \]

      14. lower-*.f64 73.1

          \[\leadsto \frac{\ell}{k \cdot t} \cdot \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot k}} \]

   10. Applied egg-rr 73.1%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k}} \]

   if 2.00000000000000006e-161 < (*.f64 l l)

   1. Initial program 53.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   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. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{2 \cdot {t}^{3}}{{\ell}^{2}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{2 \cdot {t}^{3}}{{\ell}^{2}}}} \]

      10. unpow3 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{2 \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}}{{\ell}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{2 \cdot \left(\color{blue}{{t}^{2}} \cdot t\right)}{{\ell}^{2}}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right)} \cdot t}{{\ell}^{2}}} \]

      15. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t}{{\ell}^{2}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t}{{\ell}^{2}}} \]

      17. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

      18. lower-*.f64 54.9

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

   5. Simplified 54.9%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\ell \cdot \ell}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      8. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {t}^{2}\right)}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

      12. lower-*.f64 57.6

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   8. Simplified 57.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

      8. lower-/.f64 59.2

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \cdot \ell \]

   10. Applied egg-rr 59.2%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \cdot \ell \]

       3. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

       6. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \cdot \ell \]

       7. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t}} \cdot \ell \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t} \cdot \ell \]

       9. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right)} \cdot t} \cdot \ell \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\left(\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot t\right) \cdot t} \cdot \ell \]

       11. *-commutative N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot t} \cdot \ell \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot t} \cdot \ell \]

       13. lower-*.f64 63.3

           \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right) \cdot t}} \cdot \ell \]

       14. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot t} \cdot \ell \]

       15. *-commutative N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right)} \cdot t} \cdot \ell \]

       16. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\left(\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot t\right) \cdot t} \cdot \ell \]

       17. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\left(\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot t\right) \cdot t} \cdot \ell \]

       18. associate-*l* N/A

           \[\leadsto \frac{\ell}{\left(\color{blue}{\left(k \cdot \left(k \cdot t\right)\right)} \cdot t\right) \cdot t} \cdot \ell \]

       19. associate-*l* N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(k \cdot t\right) \cdot t\right)\right)} \cdot t} \cdot \ell \]

       20. lower-*.f64 N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(k \cdot t\right) \cdot t\right)\right)} \cdot t} \cdot \ell \]

       21. lower-*.f64 N/A

           \[\leadsto \frac{\ell}{\left(k \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right) \cdot t} \cdot \ell \]

       22. lower-*.f64 67.4

           \[\leadsto \frac{\ell}{\left(k \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot t\right)\right) \cdot t} \cdot \ell \]

   12. Applied egg-rr 67.4%

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(k \cdot t\right) \cdot t\right)\right) \cdot t}} \cdot \ell \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-161}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\ell}{k \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 65.2% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* l (/ l (* t_m (* k (* t_m (* t_m k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (t_m * (k * (t_m * (t_m * k))))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (l * (l / (t_m * (k * (t_m * (t_m * k))))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (t_m * (k * (t_m * (t_m * k))))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * (l / (t_m * (k * (t_m * (t_m * k))))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(l / Float64(t_m * Float64(k * Float64(t_m * Float64(t_m * k)))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (l * (l / (t_m * (k * (t_m * (t_m * k))))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(l * N[(l / N[(t$95$m * N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
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. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

   6. unpow2 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \]

   8. associate-*r/ N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{2 \cdot {t}^{3}}{{\ell}^{2}}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{2 \cdot {t}^{3}}{{\ell}^{2}}}} \]

   10. unpow3 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{2 \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}}{{\ell}^{2}}} \]

   11. unpow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{2 \cdot \left(\color{blue}{{t}^{2}} \cdot t\right)}{{\ell}^{2}}} \]

   12. associate-*r* N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right)} \cdot t}{{\ell}^{2}}} \]

   15. unpow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t}{{\ell}^{2}}} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t}{{\ell}^{2}}} \]

   17. unpow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

   18. lower-*.f64 52.6

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

5. Simplified 52.6%

   \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\ell \cdot \ell}}} \]

6. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

   8. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {t}^{2}\right)}\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   12. lower-*.f64 58.2

       \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

8. Simplified 58.2%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

   5. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

   8. lower-/.f64 61.2

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \cdot \ell \]

10. Applied egg-rr 61.2%

    \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \cdot \ell \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \cdot \ell \]

    3. associate-*r* N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

    4. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

    5. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

    6. *-commutative N/A

       \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \cdot \ell \]

    7. associate-*r* N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t}} \cdot \ell \]

    8. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t} \cdot \ell \]

    9. associate-*l* N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right)} \cdot t} \cdot \ell \]

    10. lift-*.f64 N/A

        \[\leadsto \frac{\ell}{\left(\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot t\right) \cdot t} \cdot \ell \]

    11. *-commutative N/A

        \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot t} \cdot \ell \]

    12. lift-*.f64 N/A

        \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot t} \cdot \ell \]

    13. lower-*.f64 63.1

        \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right) \cdot t}} \cdot \ell \]

    14. lift-*.f64 N/A

        \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot t} \cdot \ell \]

    15. *-commutative N/A

        \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right)} \cdot t} \cdot \ell \]

    16. lift-*.f64 N/A

        \[\leadsto \frac{\ell}{\left(\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot t\right) \cdot t} \cdot \ell \]

    17. lift-*.f64 N/A

        \[\leadsto \frac{\ell}{\left(\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot t\right) \cdot t} \cdot \ell \]

    18. associate-*l* N/A

        \[\leadsto \frac{\ell}{\left(\color{blue}{\left(k \cdot \left(k \cdot t\right)\right)} \cdot t\right) \cdot t} \cdot \ell \]

    19. associate-*l* N/A

        \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(k \cdot t\right) \cdot t\right)\right)} \cdot t} \cdot \ell \]

    20. lower-*.f64 N/A

        \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(k \cdot t\right) \cdot t\right)\right)} \cdot t} \cdot \ell \]

    21. lower-*.f64 N/A

        \[\leadsto \frac{\ell}{\left(k \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right) \cdot t} \cdot \ell \]

    22. lower-*.f64 67.7

        \[\leadsto \frac{\ell}{\left(k \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot t\right)\right) \cdot t} \cdot \ell \]

12. Applied egg-rr 67.7%

    \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(k \cdot t\right) \cdot t\right)\right) \cdot t}} \cdot \ell \]

13. Final simplification 67.7%

    \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)} \]
14. Add Preprocessing

Alternative 19: 62.7% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{\left(t\_m \cdot k\right) \cdot \left(k \cdot \left(t\_m \cdot t\_m\right)\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* l (/ l (* (* t_m k) (* k (* t_m t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / ((t_m * k) * (k * (t_m * t_m)))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (l * (l / ((t_m * k) * (k * (t_m * t_m)))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / ((t_m * k) * (k * (t_m * t_m)))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * (l / ((t_m * k) * (k * (t_m * t_m)))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(l / Float64(Float64(t_m * k) * Float64(k * Float64(t_m * t_m))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (l * (l / ((t_m * k) * (k * (t_m * t_m)))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(l * N[(l / N[(N[(t$95$m * k), $MachinePrecision] * N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
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. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

   6. unpow2 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \]

   8. associate-*r/ N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{2 \cdot {t}^{3}}{{\ell}^{2}}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{2 \cdot {t}^{3}}{{\ell}^{2}}}} \]

   10. unpow3 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{2 \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}}{{\ell}^{2}}} \]

   11. unpow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{2 \cdot \left(\color{blue}{{t}^{2}} \cdot t\right)}{{\ell}^{2}}} \]

   12. associate-*r* N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right)} \cdot t}{{\ell}^{2}}} \]

   15. unpow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t}{{\ell}^{2}}} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t}{{\ell}^{2}}} \]

   17. unpow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

   18. lower-*.f64 52.6

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

5. Simplified 52.6%

   \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\ell \cdot \ell}}} \]

6. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

   8. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {t}^{2}\right)}\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   12. lower-*.f64 58.2

       \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

8. Simplified 58.2%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

   5. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

   8. lower-/.f64 61.2

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \cdot \ell \]

10. Applied egg-rr 61.2%

    \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \cdot \ell \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \cdot \ell \]

    3. *-commutative N/A

       \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

    4. associate-*r* N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right) \cdot k}} \cdot \ell \]

    5. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right) \cdot k} \cdot \ell \]

    6. associate-*r* N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(t \cdot t\right)\right)} \cdot k} \cdot \ell \]

    7. associate-*l* N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot k\right)}} \cdot \ell \]

    8. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot k\right)}} \cdot \ell \]

    9. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(\left(t \cdot t\right) \cdot k\right)} \cdot \ell \]

    10. lower-*.f64 64.3

        \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}} \cdot \ell \]

12. Applied egg-rr 64.3%

    \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot k\right)}} \cdot \ell \]

13. Final simplification 64.3%

    \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]
14. Add Preprocessing

Alternative 20: 62.3% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \left(t\_m \cdot k\right)\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* l (/ l (* k (* (* t_m t_m) (* t_m k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (k * ((t_m * t_m) * (t_m * k)))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (l * (l / (k * ((t_m * t_m) * (t_m * k)))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (k * ((t_m * t_m) * (t_m * k)))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * (l / (k * ((t_m * t_m) * (t_m * k)))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(l / Float64(k * Float64(Float64(t_m * t_m) * Float64(t_m * k))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (l * (l / (k * ((t_m * t_m) * (t_m * k)))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(l * N[(l / N[(k * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
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. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

   6. unpow2 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \]

   8. associate-*r/ N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{2 \cdot {t}^{3}}{{\ell}^{2}}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{2 \cdot {t}^{3}}{{\ell}^{2}}}} \]

   10. unpow3 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{2 \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}}{{\ell}^{2}}} \]

   11. unpow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{2 \cdot \left(\color{blue}{{t}^{2}} \cdot t\right)}{{\ell}^{2}}} \]

   12. associate-*r* N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right)} \cdot t}{{\ell}^{2}}} \]

   15. unpow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t}{{\ell}^{2}}} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t}{{\ell}^{2}}} \]

   17. unpow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

   18. lower-*.f64 52.6

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

5. Simplified 52.6%

   \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\ell \cdot \ell}}} \]

6. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

   8. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {t}^{2}\right)}\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   12. lower-*.f64 58.2

       \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

8. Simplified 58.2%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

   5. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

   8. lower-/.f64 61.2

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \cdot \ell \]

10. Applied egg-rr 61.2%

    \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \cdot \ell \]

    2. associate-*r* N/A

       \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

    3. *-commutative N/A

       \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right)}} \cdot \ell \]

    4. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right)}} \cdot \ell \]

    5. lower-*.f64 63.5

       \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right)} \cdot \ell \]

12. Applied egg-rr 63.5%

    \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right)}} \cdot \ell \]

13. Final simplification 63.5%

    \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \]
14. Add Preprocessing

Alternative 21: 59.3% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right)\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* l (/ l (* k (* k (* t_m (* t_m t_m))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (k * (k * (t_m * (t_m * t_m))))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (l * (l / (k * (k * (t_m * (t_m * t_m))))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (k * (k * (t_m * (t_m * t_m))))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * (l / (k * (k * (t_m * (t_m * t_m))))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(l / Float64(k * Float64(k * Float64(t_m * Float64(t_m * t_m)))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (l * (l / (k * (k * (t_m * (t_m * t_m))))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(l * N[(l / N[(k * N[(k * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
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. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

   6. unpow2 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \]

   8. associate-*r/ N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{2 \cdot {t}^{3}}{{\ell}^{2}}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{2 \cdot {t}^{3}}{{\ell}^{2}}}} \]

   10. unpow3 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{2 \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}}{{\ell}^{2}}} \]

   11. unpow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{2 \cdot \left(\color{blue}{{t}^{2}} \cdot t\right)}{{\ell}^{2}}} \]

   12. associate-*r* N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(2 \cdot {t}^{2}\right)} \cdot t}{{\ell}^{2}}} \]

   15. unpow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t}{{\ell}^{2}}} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t}{{\ell}^{2}}} \]

   17. unpow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

   18. lower-*.f64 52.6

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

5. Simplified 52.6%

   \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot t}{\ell \cdot \ell}}} \]

6. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

   8. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {t}^{2}\right)}\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   12. lower-*.f64 58.2

       \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

8. Simplified 58.2%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

   5. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

   8. lower-/.f64 61.2

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \cdot \ell \]

10. Applied egg-rr 61.2%

    \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

11. Final simplification 61.2%

    \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \]
12. Add Preprocessing

Alternative 22: 29.5% accurate, 14.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.16666666666666666}{t\_m \cdot \left(t\_m \cdot t\_m\right)} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (* (* l l) -0.16666666666666666) (* t_m (* t_m t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (((l * l) * -0.16666666666666666) / (t_m * (t_m * t_m)));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (((l * l) * (-0.16666666666666666d0)) / (t_m * (t_m * t_m)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (((l * l) * -0.16666666666666666) / (t_m * (t_m * t_m)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (((l * l) * -0.16666666666666666) / (t_m * (t_m * t_m)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(Float64(l * l) * -0.16666666666666666) / Float64(t_m * Float64(t_m * t_m))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (((l * l) * -0.16666666666666666) / (t_m * (t_m * t_m)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(N[(l * l), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] / N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{t}^{3} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2}{\color{blue}{\frac{2 \cdot \left({t}^{3} \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\frac{2 \cdot \left({t}^{3} \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]

   3. associate-*r* N/A

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(2 \cdot {t}^{3}\right) \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(2 \cdot {t}^{3}\right) \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

   5. unpow3 N/A

      \[\leadsto \frac{2}{\frac{\left(2 \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

   6. unpow2 N/A

      \[\leadsto \frac{2}{\frac{\left(2 \cdot \left(\color{blue}{{t}^{2}} \cdot t\right)\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

   7. associate-*r* N/A

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(2 \cdot {t}^{2}\right) \cdot t\right)} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(2 \cdot {t}^{2}\right) \cdot t\right)} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(\color{blue}{\left(2 \cdot {t}^{2}\right)} \cdot t\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

   10. unpow2 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot t\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

   12. lower-pow.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot \left(t \cdot t\right)\right) \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

   13. lower-sin.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot \left(t \cdot t\right)\right) \cdot t\right) \cdot {\color{blue}{\sin k}}^{2}}{{\ell}^{2} \cdot \cos k}} \]

   14. unpow2 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot \left(t \cdot t\right)\right) \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}} \]

   15. associate-*l* N/A

       \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot \left(t \cdot t\right)\right) \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot \left(t \cdot t\right)\right) \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

   17. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot \left(t \cdot t\right)\right) \cdot t\right) \cdot {\sin k}^{2}}{\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}}} \]

   18. lower-cos.f64 54.2

       \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot \left(t \cdot t\right)\right) \cdot t\right) \cdot {\sin k}^{2}}{\ell \cdot \left(\ell \cdot \color{blue}{\cos k}\right)}} \]

5. Simplified 54.2%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(2 \cdot \left(t \cdot t\right)\right) \cdot t\right) \cdot {\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

6. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{{t}^{3}} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{{t}^{3}} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{{t}^{3}}\right) + \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

8. Simplified 18.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(k \cdot k, \frac{\left(\ell \cdot \ell\right) \cdot -0.16666666666666666}{t \cdot \left(t \cdot t\right)}, \frac{\ell \cdot \ell}{t \cdot \left(t \cdot t\right)}\right)}{k \cdot k}} \]

9. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\frac{-1}{6} \cdot \frac{{\ell}^{2}}{{t}^{3}}} \]
10. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \color{blue}{\frac{\frac{-1}{6} \cdot {\ell}^{2}}{{t}^{3}}} \]

    2. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{-1}{6} \cdot {\ell}^{2}}{{t}^{3}}} \]

    3. lower-*.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{-1}{6} \cdot {\ell}^{2}}}{{t}^{3}} \]

    4. unpow2 N/A

       \[\leadsto \frac{\frac{-1}{6} \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{t}^{3}} \]

    5. lower-*.f64 N/A

       \[\leadsto \frac{\frac{-1}{6} \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{t}^{3}} \]

    6. cube-mult N/A

       \[\leadsto \frac{\frac{-1}{6} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot \left(t \cdot t\right)}} \]

    7. unpow2 N/A

       \[\leadsto \frac{\frac{-1}{6} \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{{t}^{2}}} \]

    8. lower-*.f64 N/A

       \[\leadsto \frac{\frac{-1}{6} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {t}^{2}}} \]

    9. unpow2 N/A

       \[\leadsto \frac{\frac{-1}{6} \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left(t \cdot t\right)}} \]

    10. lower-*.f64 30.1

        \[\leadsto \frac{-0.16666666666666666 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left(t \cdot t\right)}} \]

11. Simplified 30.1%

    \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot t\right)}} \]

12. Final simplification 30.1%

    \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.16666666666666666}{t \cdot \left(t \cdot t\right)} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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))))