Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.4% → 83.7%

Time: 16.4s

Alternatives: 18

Speedup: 12.5×

• 26.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.4% 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: 83.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.15 \cdot 10^{-146}:\\ \;\;\;\;\frac{\cos k}{\left(\left({\sin k}^{2} \cdot t\_m\right) \cdot k\right) \cdot k} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot \mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right)\right) \cdot \left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \sin k\right)\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.15e-146)
    (* (/ (cos k) (* (* (* (pow (sin k) 2.0) t_m) k) k)) (* (* l l) 2.0))
    (/
     2.0
     (*
      (*
       (* (tan k) (fma (/ k (* t_m t_m)) k 2.0))
       (* (* (/ t_m l) t_m) (sin k)))
      (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.15e-146) {
		tmp = (cos(k) / (((pow(sin(k), 2.0) * t_m) * k) * k)) * ((l * l) * 2.0);
	} else {
		tmp = 2.0 / (((tan(k) * fma((k / (t_m * t_m)), k, 2.0)) * (((t_m / l) * t_m) * sin(k))) * (t_m / l));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.15e-146)
		tmp = Float64(Float64(cos(k) / Float64(Float64(Float64((sin(k) ^ 2.0) * t_m) * k) * k)) * Float64(Float64(l * l) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * fma(Float64(k / Float64(t_m * t_m)), k, 2.0)) * Float64(Float64(Float64(t_m / l) * t_m) * sin(k))) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.15e-146], N[(N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.15e-146

   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

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 62.9

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

   5. Applied rewrites 62.9%

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

   if 1.15e-146 < t

   1. Initial program 70.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 88.6

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

   4. Applied rewrites 88.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 88.5

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

   6. Applied rewrites 88.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-pow N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. unpow-prod-down N/A

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

      13. pow2 N/A

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

      14. sqr-pow N/A

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

      15. pow2 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. frac-times N/A

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

      19. metadata-eval N/A

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

      20. lift-*.f64 N/A

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

      21. div-inv N/A

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

   8. Applied rewrites 88.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   10. Applied rewrites 90.7%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.15 \cdot 10^{-146}:\\ \;\;\;\;\frac{\cos k}{\left(\left({\sin k}^{2} \cdot t\right) \cdot k\right) \cdot k} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot \mathsf{fma}\left(\frac{k}{t \cdot t}, k, 2\right)\right) \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \sin k\right)\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1\right) \cdot \left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \leq 10^{+126}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k} \cdot \frac{\ell}{k \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\left(k \cdot k\right) \cdot t\_m}}{t\_m} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0)
         (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))))
       1e+126)
    (* (/ (/ l (* t_m t_m)) k) (/ l (* k t_m)))
    (* (/ (/ l (* (* k k) t_m)) t_m) (/ l t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((2.0 / (((pow((k / t_m), 2.0) + 1.0) + 1.0) * (((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)))) <= 1e+126) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	} else {
		tmp = ((l / ((k * k) * t_m)) / t_m) * (l / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((2.0d0 / (((((k / t_m) ** 2.0d0) + 1.0d0) + 1.0d0) * ((((t_m ** 3.0d0) / (l * l)) * sin(k)) * tan(k)))) <= 1d+126) then
        tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m))
    else
        tmp = ((l / ((k * k) * t_m)) / t_m) * (l / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((2.0 / (((Math.pow((k / t_m), 2.0) + 1.0) + 1.0) * (((Math.pow(t_m, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)))) <= 1e+126) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	} else {
		tmp = ((l / ((k * k) * t_m)) / t_m) * (l / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (2.0 / (((math.pow((k / t_m), 2.0) + 1.0) + 1.0) * (((math.pow(t_m, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)))) <= 1e+126:
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m))
	else:
		tmp = ((l / ((k * k) * t_m)) / t_m) * (l / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0) * Float64(Float64(Float64((t_m ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)))) <= 1e+126)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / k) * Float64(l / Float64(k * t_m)));
	else
		tmp = Float64(Float64(Float64(l / Float64(Float64(k * k) * t_m)) / t_m) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((2.0 / (((((k / t_m) ^ 2.0) + 1.0) + 1.0) * ((((t_m ^ 3.0) / (l * l)) * sin(k)) * tan(k)))) <= 1e+126)
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	else
		tmp = ((l / ((k * k) * t_m)) / t_m) * (l / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+126], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 9.99999999999999925e125

   1. Initial program 83.8%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 64.3

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

   5. Applied rewrites 64.3%

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

   7. Applied rewrites 83.8%

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

   if 9.99999999999999925e125 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 23.1%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 39.4

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

   5. Applied rewrites 39.4%

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

   7. Applied rewrites 55.1%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \leq 10^{+126}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot t}}{k} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\left(k \cdot k\right) \cdot t}}{t} \cdot \frac{\ell}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1\right) \cdot \left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \leq 10^{+126}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k} \cdot \frac{\ell}{k \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot t\_m} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0)
         (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))))
       1e+126)
    (* (/ (/ l (* t_m t_m)) k) (/ l (* k t_m)))
    (* (/ l (* (* (* k k) t_m) t_m)) (/ l t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((2.0 / (((pow((k / t_m), 2.0) + 1.0) + 1.0) * (((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)))) <= 1e+126) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	} else {
		tmp = (l / (((k * k) * t_m) * t_m)) * (l / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((2.0d0 / (((((k / t_m) ** 2.0d0) + 1.0d0) + 1.0d0) * ((((t_m ** 3.0d0) / (l * l)) * sin(k)) * tan(k)))) <= 1d+126) then
        tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m))
    else
        tmp = (l / (((k * k) * t_m) * t_m)) * (l / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((2.0 / (((Math.pow((k / t_m), 2.0) + 1.0) + 1.0) * (((Math.pow(t_m, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)))) <= 1e+126) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	} else {
		tmp = (l / (((k * k) * t_m) * t_m)) * (l / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (2.0 / (((math.pow((k / t_m), 2.0) + 1.0) + 1.0) * (((math.pow(t_m, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)))) <= 1e+126:
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m))
	else:
		tmp = (l / (((k * k) * t_m) * t_m)) * (l / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0) * Float64(Float64(Float64((t_m ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)))) <= 1e+126)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / k) * Float64(l / Float64(k * t_m)));
	else
		tmp = Float64(Float64(l / Float64(Float64(Float64(k * k) * t_m) * t_m)) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((2.0 / (((((k / t_m) ^ 2.0) + 1.0) + 1.0) * ((((t_m ^ 3.0) / (l * l)) * sin(k)) * tan(k)))) <= 1e+126)
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	else
		tmp = (l / (((k * k) * t_m) * t_m)) * (l / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+126], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 9.99999999999999925e125

   1. Initial program 83.8%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 64.3

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

   5. Applied rewrites 64.3%

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

   7. Applied rewrites 83.8%

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

   if 9.99999999999999925e125 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 23.1%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 39.4

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

   5. Applied rewrites 39.4%

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

   7. Applied rewrites 54.2%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \leq 10^{+126}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot t}}{k} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t} \cdot \frac{\ell}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1\right) \cdot \left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \leq 10^{+126}:\\ \;\;\;\;\frac{\ell}{\left(t\_m \cdot t\_m\right) \cdot k} \cdot \frac{\ell}{k \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot t\_m} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0)
         (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))))
       1e+126)
    (* (/ l (* (* t_m t_m) k)) (/ l (* k t_m)))
    (* (/ l (* (* (* k k) t_m) t_m)) (/ l t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((2.0 / (((pow((k / t_m), 2.0) + 1.0) + 1.0) * (((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)))) <= 1e+126) {
		tmp = (l / ((t_m * t_m) * k)) * (l / (k * t_m));
	} else {
		tmp = (l / (((k * k) * t_m) * t_m)) * (l / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((2.0d0 / (((((k / t_m) ** 2.0d0) + 1.0d0) + 1.0d0) * ((((t_m ** 3.0d0) / (l * l)) * sin(k)) * tan(k)))) <= 1d+126) then
        tmp = (l / ((t_m * t_m) * k)) * (l / (k * t_m))
    else
        tmp = (l / (((k * k) * t_m) * t_m)) * (l / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((2.0 / (((Math.pow((k / t_m), 2.0) + 1.0) + 1.0) * (((Math.pow(t_m, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)))) <= 1e+126) {
		tmp = (l / ((t_m * t_m) * k)) * (l / (k * t_m));
	} else {
		tmp = (l / (((k * k) * t_m) * t_m)) * (l / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (2.0 / (((math.pow((k / t_m), 2.0) + 1.0) + 1.0) * (((math.pow(t_m, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)))) <= 1e+126:
		tmp = (l / ((t_m * t_m) * k)) * (l / (k * t_m))
	else:
		tmp = (l / (((k * k) * t_m) * t_m)) * (l / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0) * Float64(Float64(Float64((t_m ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)))) <= 1e+126)
		tmp = Float64(Float64(l / Float64(Float64(t_m * t_m) * k)) * Float64(l / Float64(k * t_m)));
	else
		tmp = Float64(Float64(l / Float64(Float64(Float64(k * k) * t_m) * t_m)) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((2.0 / (((((k / t_m) ^ 2.0) + 1.0) + 1.0) * ((((t_m ^ 3.0) / (l * l)) * sin(k)) * tan(k)))) <= 1e+126)
		tmp = (l / ((t_m * t_m) * k)) * (l / (k * t_m));
	else
		tmp = (l / (((k * k) * t_m) * t_m)) * (l / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+126], N[(N[(l / N[(N[(t$95$m * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 9.99999999999999925e125

   1. Initial program 83.8%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 64.3

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

   5. Applied rewrites 64.3%

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

   7. Applied rewrites 83.8%

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

   if 9.99999999999999925e125 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 23.1%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 39.4

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

   5. Applied rewrites 39.4%

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

   7. Applied rewrites 54.2%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \leq 10^{+126}:\\ \;\;\;\;\frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t} \cdot \frac{\ell}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3 \cdot 10^{-91}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{t\_m \cdot t\_m}{\ell} \cdot t\_m, 2, \left(\frac{k}{\ell} \cdot k\right) \cdot \left(\mathsf{fma}\left(t\_m \cdot t\_m, 0.3333333333333333, 1\right) \cdot t\_m\right)\right)}{\ell}}\\ \mathbf{elif}\;t\_m \leq 2.3 \cdot 10^{+145}:\\ \;\;\;\;\frac{\ell}{\tan k \cdot \mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right)} \cdot \frac{2}{\left(\frac{\sin k}{\ell} \cdot t\_m\right) \cdot \left(t\_m \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right) \cdot \left(\left(\left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3e-91)
    (/
     2.0
     (/
      (*
       (* k k)
       (fma
        (* (/ (* t_m t_m) l) t_m)
        2.0
        (* (* (/ k l) k) (* (fma (* t_m t_m) 0.3333333333333333 1.0) t_m))))
      l))
    (if (<= t_m 2.3e+145)
      (*
       (/ l (* (tan k) (fma (/ k (* t_m t_m)) k 2.0)))
       (/ 2.0 (* (* (/ (sin k) l) t_m) (* t_m t_m))))
      (/
       2.0
       (*
        (fma (/ k t_m) (/ k t_m) 2.0)
        (* (* (* (* (/ t_m l) t_m) (/ t_m l)) (sin k)) (tan k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3e-91) {
		tmp = 2.0 / (((k * k) * fma((((t_m * t_m) / l) * t_m), 2.0, (((k / l) * k) * (fma((t_m * t_m), 0.3333333333333333, 1.0) * t_m)))) / l);
	} else if (t_m <= 2.3e+145) {
		tmp = (l / (tan(k) * fma((k / (t_m * t_m)), k, 2.0))) * (2.0 / (((sin(k) / l) * t_m) * (t_m * t_m)));
	} else {
		tmp = 2.0 / (fma((k / t_m), (k / t_m), 2.0) * (((((t_m / l) * t_m) * (t_m / l)) * sin(k)) * tan(k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3e-91)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * fma(Float64(Float64(Float64(t_m * t_m) / l) * t_m), 2.0, Float64(Float64(Float64(k / l) * k) * Float64(fma(Float64(t_m * t_m), 0.3333333333333333, 1.0) * t_m)))) / l));
	elseif (t_m <= 2.3e+145)
		tmp = Float64(Float64(l / Float64(tan(k) * fma(Float64(k / Float64(t_m * t_m)), k, 2.0))) * Float64(2.0 / Float64(Float64(Float64(sin(k) / l) * t_m) * Float64(t_m * t_m))));
	else
		tmp = Float64(2.0 / Float64(fma(Float64(k / t_m), Float64(k / t_m), 2.0) * Float64(Float64(Float64(Float64(Float64(t_m / l) * t_m) * Float64(t_m / l)) * sin(k)) * tan(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, 3e-91], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0 + N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.3e+145], N[(N[(l / N[(N[Tan[k], $MachinePrecision] * N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.0000000000000002e-91

   1. Initial program 52.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 51.8%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 64.1%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 62.9%

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

   if 3.0000000000000002e-91 < t < 2.3e145

   1. Initial program 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 87.4

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

   4. Applied rewrites 87.4%

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

   6. Applied rewrites 92.7%

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

   7. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 94.7

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

   9. Applied rewrites 94.7%

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

   if 2.3e145 < t

   1. Initial program 80.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 92.6

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

   4. Applied rewrites 92.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 92.6

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

   6. Applied rewrites 92.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-pow N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. unpow-prod-down N/A

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

      13. pow2 N/A

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

      14. sqr-pow N/A

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

      15. pow2 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. frac-times N/A

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

      19. metadata-eval N/A

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

      20. lift-*.f64 N/A

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

      21. div-inv N/A

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

   8. Applied rewrites 92.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. lift-fma.f64 92.6

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

   10. Applied rewrites 92.6%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-91}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot t, 2, \left(\frac{k}{\ell} \cdot k\right) \cdot \left(\mathsf{fma}\left(t \cdot t, 0.3333333333333333, 1\right) \cdot t\right)\right)}{\ell}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+145}:\\ \;\;\;\;\frac{\ell}{\tan k \cdot \mathsf{fma}\left(\frac{k}{t \cdot t}, k, 2\right)} \cdot \frac{2}{\left(\frac{\sin k}{\ell} \cdot t\right) \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3 \cdot 10^{-91}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{t\_m \cdot t\_m}{\ell} \cdot t\_m, 2, \left(\frac{k}{\ell} \cdot k\right) \cdot \left(\mathsf{fma}\left(t\_m \cdot t\_m, 0.3333333333333333, 1\right) \cdot t\_m\right)\right)}{\ell}}\\ \mathbf{elif}\;t\_m \leq 4.3 \cdot 10^{+158}:\\ \;\;\;\;\frac{\ell}{\tan k \cdot \mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right)} \cdot \frac{2}{\left(\frac{\sin k}{\ell} \cdot t\_m\right) \cdot \left(t\_m \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot t\_m} \cdot \left({\left(\frac{k \cdot t\_m}{\ell}\right)}^{-1} \cdot \frac{\ell}{t\_m}\right)\\ \end{array} \end{array} \]

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 3e-91)
    (/
     2.0
     (/
      (*
       (* k k)
       (fma
        (* (/ (* t_m t_m) l) t_m)
        2.0
        (* (* (/ k l) k) (* (fma (* t_m t_m) 0.3333333333333333 1.0) t_m))))
      l))
    (if (<= t_m 4.3e+158)
      (*
       (/ l (* (tan k) (fma (/ k (* t_m t_m)) k 2.0)))
       (/ 2.0 (* (* (/ (sin k) l) t_m) (* t_m t_m))))
      (* (/ 1.0 (* k t_m)) (* (pow (/ (* k t_m) l) -1.0) (/ l t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3e-91) {
		tmp = 2.0 / (((k * k) * fma((((t_m * t_m) / l) * t_m), 2.0, (((k / l) * k) * (fma((t_m * t_m), 0.3333333333333333, 1.0) * t_m)))) / l);
	} else if (t_m <= 4.3e+158) {
		tmp = (l / (tan(k) * fma((k / (t_m * t_m)), k, 2.0))) * (2.0 / (((sin(k) / l) * t_m) * (t_m * t_m)));
	} else {
		tmp = (1.0 / (k * t_m)) * (pow(((k * t_m) / l), -1.0) * (l / t_m));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3e-91)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * fma(Float64(Float64(Float64(t_m * t_m) / l) * t_m), 2.0, Float64(Float64(Float64(k / l) * k) * Float64(fma(Float64(t_m * t_m), 0.3333333333333333, 1.0) * t_m)))) / l));
	elseif (t_m <= 4.3e+158)
		tmp = Float64(Float64(l / Float64(tan(k) * fma(Float64(k / Float64(t_m * t_m)), k, 2.0))) * Float64(2.0 / Float64(Float64(Float64(sin(k) / l) * t_m) * Float64(t_m * t_m))));
	else
		tmp = Float64(Float64(1.0 / Float64(k * t_m)) * Float64((Float64(Float64(k * t_m) / l) ^ -1.0) * Float64(l / t_m)));
	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, 3e-91], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0 + N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.3e+158], N[(N[(l / N[(N[Tan[k], $MachinePrecision] * N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision], -1.0], $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.0000000000000002e-91

   1. Initial program 52.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 51.8%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 64.1%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 62.9%

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

   if 3.0000000000000002e-91 < t < 4.3e158

   1. Initial program 71.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 87.8

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

   4. Applied rewrites 87.8%

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

   6. Applied rewrites 92.9%

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

   7. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 94.9

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

   9. Applied rewrites 94.9%

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

   if 4.3e158 < t

   1. Initial program 78.5%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 43.5

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

   5. Applied rewrites 43.5%

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

   7. Applied rewrites 78.5%

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

   9. Applied rewrites 99.8%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-91}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot t, 2, \left(\frac{k}{\ell} \cdot k\right) \cdot \left(\mathsf{fma}\left(t \cdot t, 0.3333333333333333, 1\right) \cdot t\right)\right)}{\ell}}\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{+158}:\\ \;\;\;\;\frac{\ell}{\tan k \cdot \mathsf{fma}\left(\frac{k}{t \cdot t}, k, 2\right)} \cdot \frac{2}{\left(\frac{\sin k}{\ell} \cdot t\right) \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot t} \cdot \left({\left(\frac{k \cdot t}{\ell}\right)}^{-1} \cdot \frac{\ell}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.3 \cdot 10^{-88}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{t\_m \cdot t\_m}{\ell} \cdot t\_m, 2, \left(\frac{k}{\ell} \cdot k\right) \cdot \left(\mathsf{fma}\left(t\_m \cdot t\_m, 0.3333333333333333, 1\right) \cdot t\_m\right)\right)}{\ell}}\\ \mathbf{elif}\;t\_m \leq 5.9 \cdot 10^{+150}:\\ \;\;\;\;\left(\frac{2}{\left(\sin k \cdot t\_m\right) \cdot t\_m} \cdot \frac{\ell}{t\_m}\right) \cdot \frac{\ell}{\tan k \cdot \mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot t\_m} \cdot \left({\left(\frac{k \cdot t\_m}{\ell}\right)}^{-1} \cdot \frac{\ell}{t\_m}\right)\\ \end{array} \end{array} \]

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 3.3e-88)
    (/
     2.0
     (/
      (*
       (* k k)
       (fma
        (* (/ (* t_m t_m) l) t_m)
        2.0
        (* (* (/ k l) k) (* (fma (* t_m t_m) 0.3333333333333333 1.0) t_m))))
      l))
    (if (<= t_m 5.9e+150)
      (*
       (* (/ 2.0 (* (* (sin k) t_m) t_m)) (/ l t_m))
       (/ l (* (tan k) (fma (/ k (* t_m t_m)) k 2.0))))
      (* (/ 1.0 (* k t_m)) (* (pow (/ (* k t_m) l) -1.0) (/ l t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.3e-88) {
		tmp = 2.0 / (((k * k) * fma((((t_m * t_m) / l) * t_m), 2.0, (((k / l) * k) * (fma((t_m * t_m), 0.3333333333333333, 1.0) * t_m)))) / l);
	} else if (t_m <= 5.9e+150) {
		tmp = ((2.0 / ((sin(k) * t_m) * t_m)) * (l / t_m)) * (l / (tan(k) * fma((k / (t_m * t_m)), k, 2.0)));
	} else {
		tmp = (1.0 / (k * t_m)) * (pow(((k * t_m) / l), -1.0) * (l / t_m));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.3e-88)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * fma(Float64(Float64(Float64(t_m * t_m) / l) * t_m), 2.0, Float64(Float64(Float64(k / l) * k) * Float64(fma(Float64(t_m * t_m), 0.3333333333333333, 1.0) * t_m)))) / l));
	elseif (t_m <= 5.9e+150)
		tmp = Float64(Float64(Float64(2.0 / Float64(Float64(sin(k) * t_m) * t_m)) * Float64(l / t_m)) * Float64(l / Float64(tan(k) * fma(Float64(k / Float64(t_m * t_m)), k, 2.0))));
	else
		tmp = Float64(Float64(1.0 / Float64(k * t_m)) * Float64((Float64(Float64(k * t_m) / l) ^ -1.0) * Float64(l / t_m)));
	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, 3.3e-88], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0 + N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.9e+150], N[(N[(N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[Tan[k], $MachinePrecision] * N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision], -1.0], $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.29999999999999994e-88

   1. Initial program 52.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 51.8%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 64.1%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 62.9%

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

   if 3.29999999999999994e-88 < t < 5.90000000000000023e150

   1. Initial program 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 87.4

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

   4. Applied rewrites 87.4%

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

   6. Applied rewrites 92.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow-prod-down N/A

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

      8. inv-pow N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. inv-pow N/A

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

      12. clear-num N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 91.4

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

   8. Applied rewrites 91.4%

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

   if 5.90000000000000023e150 < t

   1. Initial program 80.2%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 48.0

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

   5. Applied rewrites 48.0%

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

   7. Applied rewrites 80.2%

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

   9. Applied rewrites 99.8%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{-88}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot t, 2, \left(\frac{k}{\ell} \cdot k\right) \cdot \left(\mathsf{fma}\left(t \cdot t, 0.3333333333333333, 1\right) \cdot t\right)\right)}{\ell}}\\ \mathbf{elif}\;t \leq 5.9 \cdot 10^{+150}:\\ \;\;\;\;\left(\frac{2}{\left(\sin k \cdot t\right) \cdot t} \cdot \frac{\ell}{t}\right) \cdot \frac{\ell}{\tan k \cdot \mathsf{fma}\left(\frac{k}{t \cdot t}, k, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot t} \cdot \left({\left(\frac{k \cdot t}{\ell}\right)}^{-1} \cdot \frac{\ell}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-86}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{t\_m \cdot t\_m}{\ell} \cdot t\_m, 2, \left(\frac{k}{\ell} \cdot k\right) \cdot \left(\mathsf{fma}\left(t\_m \cdot t\_m, 0.3333333333333333, 1\right) \cdot t\_m\right)\right)}{\ell}}\\ \mathbf{elif}\;t\_m \leq 1.45 \cdot 10^{+141}:\\ \;\;\;\;\left(\frac{2}{\left(\left(\sin k \cdot t\_m\right) \cdot t\_m\right) \cdot t\_m} \cdot \ell\right) \cdot \frac{\ell}{\tan k \cdot \mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(\left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1e-86)
    (/
     2.0
     (/
      (*
       (* k k)
       (fma
        (* (/ (* t_m t_m) l) t_m)
        2.0
        (* (* (/ k l) k) (* (fma (* t_m t_m) 0.3333333333333333 1.0) t_m))))
      l))
    (if (<= t_m 1.45e+141)
      (*
       (* (/ 2.0 (* (* (* (sin k) t_m) t_m) t_m)) l)
       (/ l (* (tan k) (fma (/ k (* t_m t_m)) k 2.0))))
      (/
       2.0
       (* 2.0 (* (* (* (* (/ t_m l) t_m) (/ t_m l)) (sin k)) (tan k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1e-86) {
		tmp = 2.0 / (((k * k) * fma((((t_m * t_m) / l) * t_m), 2.0, (((k / l) * k) * (fma((t_m * t_m), 0.3333333333333333, 1.0) * t_m)))) / l);
	} else if (t_m <= 1.45e+141) {
		tmp = ((2.0 / (((sin(k) * t_m) * t_m) * t_m)) * l) * (l / (tan(k) * fma((k / (t_m * t_m)), k, 2.0)));
	} else {
		tmp = 2.0 / (2.0 * (((((t_m / l) * t_m) * (t_m / l)) * sin(k)) * tan(k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1e-86)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * fma(Float64(Float64(Float64(t_m * t_m) / l) * t_m), 2.0, Float64(Float64(Float64(k / l) * k) * Float64(fma(Float64(t_m * t_m), 0.3333333333333333, 1.0) * t_m)))) / l));
	elseif (t_m <= 1.45e+141)
		tmp = Float64(Float64(Float64(2.0 / Float64(Float64(Float64(sin(k) * t_m) * t_m) * t_m)) * l) * Float64(l / Float64(tan(k) * fma(Float64(k / Float64(t_m * t_m)), k, 2.0))));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(Float64(Float64(Float64(t_m / l) * t_m) * Float64(t_m / l)) * sin(k)) * tan(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, 1e-86], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0 + N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.45e+141], N[(N[(N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * N[(l / N[(N[Tan[k], $MachinePrecision] * N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.00000000000000008e-86

   1. Initial program 52.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 51.8%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 64.1%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 62.9%

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

   if 1.00000000000000008e-86 < t < 1.45000000000000003e141

   1. Initial program 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 87.4

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

   4. Applied rewrites 87.4%

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

   6. Applied rewrites 92.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 92.7

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. cube-mult N/A

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

   8. Applied rewrites 88.2%

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

   if 1.45000000000000003e141 < t

   1. Initial program 80.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 92.6

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

   4. Applied rewrites 92.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 92.6

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

   6. Applied rewrites 92.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-pow N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. unpow-prod-down N/A

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

      13. pow2 N/A

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

      14. sqr-pow N/A

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

      15. pow2 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. frac-times N/A

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

      19. metadata-eval N/A

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

      20. lift-*.f64 N/A

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

      21. div-inv N/A

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

   8. Applied rewrites 92.6%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 92.6%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-86}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot t, 2, \left(\frac{k}{\ell} \cdot k\right) \cdot \left(\mathsf{fma}\left(t \cdot t, 0.3333333333333333, 1\right) \cdot t\right)\right)}{\ell}}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+141}:\\ \;\;\;\;\left(\frac{2}{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t} \cdot \ell\right) \cdot \frac{\ell}{\tan k \cdot \mathsf{fma}\left(\frac{k}{t \cdot t}, k, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{-91}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{t\_m \cdot t\_m}{\ell} \cdot t\_m, 2, \left(\frac{k}{\ell} \cdot k\right) \cdot \left(\mathsf{fma}\left(t\_m \cdot t\_m, 0.3333333333333333, 1\right) \cdot t\_m\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot \mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right)\right) \cdot \left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \sin k\right)\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5e-91)
    (/
     2.0
     (/
      (*
       (* k k)
       (fma
        (* (/ (* t_m t_m) l) t_m)
        2.0
        (* (* (/ k l) k) (* (fma (* t_m t_m) 0.3333333333333333 1.0) t_m))))
      l))
    (/
     2.0
     (*
      (*
       (* (tan k) (fma (/ k (* t_m t_m)) k 2.0))
       (* (* (/ t_m l) t_m) (sin k)))
      (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5e-91) {
		tmp = 2.0 / (((k * k) * fma((((t_m * t_m) / l) * t_m), 2.0, (((k / l) * k) * (fma((t_m * t_m), 0.3333333333333333, 1.0) * t_m)))) / l);
	} else {
		tmp = 2.0 / (((tan(k) * fma((k / (t_m * t_m)), k, 2.0)) * (((t_m / l) * t_m) * sin(k))) * (t_m / l));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5e-91)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * fma(Float64(Float64(Float64(t_m * t_m) / l) * t_m), 2.0, Float64(Float64(Float64(k / l) * k) * Float64(fma(Float64(t_m * t_m), 0.3333333333333333, 1.0) * t_m)))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * fma(Float64(k / Float64(t_m * t_m)), k, 2.0)) * Float64(Float64(Float64(t_m / l) * t_m) * sin(k))) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5e-91], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0 + N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.99999999999999997e-91

   1. Initial program 52.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 51.8%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 64.1%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 62.9%

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

   if 4.99999999999999997e-91 < t

   1. Initial program 73.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 88.9

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

   4. Applied rewrites 88.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 88.8

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

   6. Applied rewrites 88.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-pow N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. unpow-prod-down N/A

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

      13. pow2 N/A

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

      14. sqr-pow N/A

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

      15. pow2 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. frac-times N/A

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

      19. metadata-eval N/A

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

      20. lift-*.f64 N/A

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

      21. div-inv N/A

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

   8. Applied rewrites 88.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   10. Applied rewrites 92.2%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-91}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot t, 2, \left(\frac{k}{\ell} \cdot k\right) \cdot \left(\mathsf{fma}\left(t \cdot t, 0.3333333333333333, 1\right) \cdot t\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot \mathsf{fma}\left(\frac{k}{t \cdot t}, k, 2\right)\right) \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \sin k\right)\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t\_m}}{t\_m} \cdot \frac{\frac{\ell}{k}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(\left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 2e+102)
    (* (/ (/ l (* k t_m)) t_m) (/ (/ l k) t_m))
    (/ 2.0 (* 2.0 (* (* (* (* (/ t_m l) t_m) (/ t_m l)) (sin k)) (tan k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 2e+102) {
		tmp = ((l / (k * t_m)) / t_m) * ((l / k) / t_m);
	} else {
		tmp = 2.0 / (2.0 * (((((t_m / l) * t_m) * (t_m / l)) * sin(k)) * tan(k)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 2d+102) then
        tmp = ((l / (k * t_m)) / t_m) * ((l / k) / t_m)
    else
        tmp = 2.0d0 / (2.0d0 * (((((t_m / l) * t_m) * (t_m / l)) * sin(k)) * tan(k)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 2e+102) {
		tmp = ((l / (k * t_m)) / t_m) * ((l / k) / t_m);
	} else {
		tmp = 2.0 / (2.0 * (((((t_m / l) * t_m) * (t_m / l)) * Math.sin(k)) * Math.tan(k)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 2e+102:
		tmp = ((l / (k * t_m)) / t_m) * ((l / k) / t_m)
	else:
		tmp = 2.0 / (2.0 * (((((t_m / l) * t_m) * (t_m / l)) * math.sin(k)) * math.tan(k)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 2e+102)
		tmp = Float64(Float64(Float64(l / Float64(k * t_m)) / t_m) * Float64(Float64(l / k) / t_m));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(Float64(Float64(Float64(t_m / l) * t_m) * Float64(t_m / l)) * sin(k)) * tan(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 2e+102)
		tmp = ((l / (k * t_m)) / t_m) * ((l / k) / t_m);
	else
		tmp = 2.0 / (2.0 * (((((t_m / l) * t_m) * (t_m / l)) * sin(k)) * tan(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 2e+102], N[(N[(N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.99999999999999995e102

   1. Initial program 70.5%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 55.5

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

   5. Applied rewrites 55.5%

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

   7. Applied rewrites 80.3%

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

   if 1.99999999999999995e102 < (*.f64 l l)

   1. Initial program 42.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 35.1

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

   4. Applied rewrites 35.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 35.1

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

   6. Applied rewrites 35.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-pow N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. unpow-prod-down N/A

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

      13. pow2 N/A

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

      14. sqr-pow N/A

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

      15. pow2 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. frac-times N/A

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

      19. metadata-eval N/A

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

      20. lift-*.f64 N/A

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

      21. div-inv N/A

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

   8. Applied rewrites 66.9%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 72.1%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t} \cdot \frac{\frac{\ell}{k}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.05 \cdot 10^{-50}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{t\_m \cdot t\_m}{\ell} \cdot t\_m, 2, \left(\frac{k}{\ell} \cdot k\right) \cdot \left(\mathsf{fma}\left(t\_m \cdot t\_m, 0.3333333333333333, 1\right) \cdot t\_m\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot t\_m} \cdot \left({\left(\frac{k \cdot t\_m}{\ell}\right)}^{-1} \cdot \frac{\ell}{t\_m}\right)\\ \end{array} \end{array} \]

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-50)
    (/
     2.0
     (/
      (*
       (* k k)
       (fma
        (* (/ (* t_m t_m) l) t_m)
        2.0
        (* (* (/ k l) k) (* (fma (* t_m t_m) 0.3333333333333333 1.0) t_m))))
      l))
    (* (/ 1.0 (* k t_m)) (* (pow (/ (* k t_m) l) -1.0) (/ l t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.05e-50) {
		tmp = 2.0 / (((k * k) * fma((((t_m * t_m) / l) * t_m), 2.0, (((k / l) * k) * (fma((t_m * t_m), 0.3333333333333333, 1.0) * t_m)))) / l);
	} else {
		tmp = (1.0 / (k * t_m)) * (pow(((k * t_m) / l), -1.0) * (l / t_m));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.05e-50)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * fma(Float64(Float64(Float64(t_m * t_m) / l) * t_m), 2.0, Float64(Float64(Float64(k / l) * k) * Float64(fma(Float64(t_m * t_m), 0.3333333333333333, 1.0) * t_m)))) / l));
	else
		tmp = Float64(Float64(1.0 / Float64(k * t_m)) * Float64((Float64(Float64(k * t_m) / l) ^ -1.0) * Float64(l / t_m)));
	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-50], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0 + N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision], -1.0], $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.05e-50

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 54.4%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 64.3%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 63.8%

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

   if 1.05e-50 < t

   1. Initial program 71.7%

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

   3. Taylor expanded in k around 0

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

      1. 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. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 59.3

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

   5. Applied rewrites 59.3%

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

   7. Applied rewrites 74.6%

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

   9. Applied rewrites 83.1%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-50}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot t, 2, \left(\frac{k}{\ell} \cdot k\right) \cdot \left(\mathsf{fma}\left(t \cdot t, 0.3333333333333333, 1\right) \cdot t\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot t} \cdot \left({\left(\frac{k \cdot t}{\ell}\right)}^{-1} \cdot \frac{\ell}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.05 \cdot 10^{-50}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{t\_m \cdot t\_m}{\ell} \cdot t\_m, 2, \left(\frac{k}{\ell} \cdot k\right) \cdot \left(\mathsf{fma}\left(t\_m \cdot t\_m, 0.3333333333333333, 1\right) \cdot t\_m\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k \cdot t\_m} \cdot \frac{\ell}{t\_m}\right) \cdot \frac{1}{k \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.05e-50)
    (/
     2.0
     (/
      (*
       (* k k)
       (fma
        (* (/ (* t_m t_m) l) t_m)
        2.0
        (* (* (/ k l) k) (* (fma (* t_m t_m) 0.3333333333333333 1.0) t_m))))
      l))
    (* (* (/ l (* k t_m)) (/ l t_m)) (/ 1.0 (* k t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.05e-50) {
		tmp = 2.0 / (((k * k) * fma((((t_m * t_m) / l) * t_m), 2.0, (((k / l) * k) * (fma((t_m * t_m), 0.3333333333333333, 1.0) * t_m)))) / l);
	} else {
		tmp = ((l / (k * t_m)) * (l / t_m)) * (1.0 / (k * t_m));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.05e-50)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * fma(Float64(Float64(Float64(t_m * t_m) / l) * t_m), 2.0, Float64(Float64(Float64(k / l) * k) * Float64(fma(Float64(t_m * t_m), 0.3333333333333333, 1.0) * t_m)))) / l));
	else
		tmp = Float64(Float64(Float64(l / Float64(k * t_m)) * Float64(l / t_m)) * Float64(1.0 / Float64(k * t_m)));
	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-50], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0 + N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.05e-50

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 54.4%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 64.3%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 63.8%

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

   if 1.05e-50 < t

   1. Initial program 71.7%

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

   3. Taylor expanded in k around 0

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

      1. 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. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 59.3

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

   5. Applied rewrites 59.3%

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

   7. Applied rewrites 74.6%

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

   9. Applied rewrites 83.0%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-50}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot t, 2, \left(\frac{k}{\ell} \cdot k\right) \cdot \left(\mathsf{fma}\left(t \cdot t, 0.3333333333333333, 1\right) \cdot t\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}\right) \cdot \frac{1}{k \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.6 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\left(k \cdot k\right) \cdot t\_m} \cdot \ell}{t\_m}}{t\_m}\\ \mathbf{elif}\;t\_m \leq 1.35 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right)\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k \cdot t\_m} \cdot \frac{\ell}{t\_m}\right) \cdot \frac{1}{k \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.6e-153)
    (/ (/ (* (/ l (* (* k k) t_m)) l) t_m) t_m)
    (if (<= t_m 1.35e-14)
      (/
       2.0
       (*
        (* (* (/ k l) k) (* (* t_m t_m) (/ t_m l)))
        (fma (/ k t_m) (/ k t_m) 2.0)))
      (* (* (/ l (* k t_m)) (/ l t_m)) (/ 1.0 (* k t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.6e-153) {
		tmp = (((l / ((k * k) * t_m)) * l) / t_m) / t_m;
	} else if (t_m <= 1.35e-14) {
		tmp = 2.0 / ((((k / l) * k) * ((t_m * t_m) * (t_m / l))) * fma((k / t_m), (k / t_m), 2.0));
	} else {
		tmp = ((l / (k * t_m)) * (l / t_m)) * (1.0 / (k * t_m));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.6e-153)
		tmp = Float64(Float64(Float64(Float64(l / Float64(Float64(k * k) * t_m)) * l) / t_m) / t_m);
	elseif (t_m <= 1.35e-14)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * k) * Float64(Float64(t_m * t_m) * Float64(t_m / l))) * fma(Float64(k / t_m), Float64(k / t_m), 2.0)));
	else
		tmp = Float64(Float64(Float64(l / Float64(k * t_m)) * Float64(l / t_m)) * Float64(1.0 / Float64(k * t_m)));
	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, 1.6e-153], N[(N[(N[(N[(l / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision], If[LessEqual[t$95$m, 1.35e-14], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.6e-153

   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 k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 52.2

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

   5. Applied rewrites 52.2%

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

   7. Applied rewrites 61.3%

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

   if 1.6e-153 < t < 1.3499999999999999e-14

   1. Initial program 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow3 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 76.2

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

   4. Applied rewrites 76.2%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 83.9%

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left(\left(\mathsf{fma}\left(\frac{k \cdot k}{\ell}, 0.16666666666666666, \frac{1}{\ell}\right) \cdot k\right) \cdot k\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 83.9

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\left(\mathsf{fma}\left(\frac{k \cdot k}{\ell}, 0.16666666666666666, \frac{1}{\ell}\right) \cdot k\right) \cdot k\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   9. Applied rewrites 83.9%

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\left(\mathsf{fma}\left(\frac{k \cdot k}{\ell}, 0.16666666666666666, \frac{1}{\ell}\right) \cdot k\right) \cdot k\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   10. Taylor expanded in k around 0

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

       1. *-lft-identity N/A

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

       2. associate-*l/ N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. associate-*l/ N/A

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

       6. *-lft-identity N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 83.9

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

   12. Applied rewrites 83.9%

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

   if 1.3499999999999999e-14 < t

   1. Initial program 72.1%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 58.7

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

   5. Applied rewrites 58.7%

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

   7. Applied rewrites 75.3%

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

   9. Applied rewrites 83.0%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \ell}{t}}{t}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}\right) \cdot \frac{1}{k \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 71.7% accurate, 7.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10^{+26}:\\ \;\;\;\;\left(\frac{\ell}{k \cdot t\_m} \cdot \frac{\ell}{t\_m}\right) \cdot \frac{1}{k \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\left(k \cdot k\right) \cdot t\_m} \cdot \ell}{t\_m}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1e+26)
    (* (* (/ l (* k t_m)) (/ l t_m)) (/ 1.0 (* k t_m)))
    (/ (/ (* (/ l (* (* k k) t_m)) l) t_m) t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1e+26) {
		tmp = ((l / (k * t_m)) * (l / t_m)) * (1.0 / (k * t_m));
	} else {
		tmp = (((l / ((k * k) * t_m)) * l) / t_m) / t_m;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1d+26) then
        tmp = ((l / (k * t_m)) * (l / t_m)) * (1.0d0 / (k * t_m))
    else
        tmp = (((l / ((k * k) * t_m)) * l) / t_m) / t_m
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1e+26) {
		tmp = ((l / (k * t_m)) * (l / t_m)) * (1.0 / (k * t_m));
	} else {
		tmp = (((l / ((k * k) * t_m)) * l) / t_m) / t_m;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1e+26:
		tmp = ((l / (k * t_m)) * (l / t_m)) * (1.0 / (k * t_m))
	else:
		tmp = (((l / ((k * k) * t_m)) * l) / t_m) / t_m
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1e+26)
		tmp = Float64(Float64(Float64(l / Float64(k * t_m)) * Float64(l / t_m)) * Float64(1.0 / Float64(k * t_m)));
	else
		tmp = Float64(Float64(Float64(Float64(l / Float64(Float64(k * k) * t_m)) * l) / t_m) / t_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1e+26)
		tmp = ((l / (k * t_m)) * (l / t_m)) * (1.0 / (k * t_m));
	else
		tmp = (((l / ((k * k) * t_m)) * l) / t_m) / t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1e+26], N[(N[(N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.00000000000000005e26

   1. Initial program 62.4%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 55.5

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

   5. Applied rewrites 55.5%

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

   7. Applied rewrites 68.6%

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

   9. Applied rewrites 76.7%

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

   if 1.00000000000000005e26 < k

   1. Initial program 43.0%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 47.7

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   5. Applied rewrites 47.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.9%

      \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t}}{t}}{\color{blue}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{+26}:\\ \;\;\;\;\left(\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}\right) \cdot \frac{1}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \ell}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 70.8% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.65 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\left(k \cdot k\right) \cdot t\_m} \cdot \ell}{t\_m}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(k \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)\right) \cdot \frac{t\_m}{\ell}} \cdot \ell\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.65e+80)
    (/ (/ (* (/ l (* (* k k) t_m)) l) t_m) t_m)
    (* (/ 1.0 (* (* (* k t_m) (* k t_m)) (/ t_m l))) l))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.65e+80) {
		tmp = (((l / ((k * k) * t_m)) * l) / t_m) / t_m;
	} else {
		tmp = (1.0 / (((k * t_m) * (k * t_m)) * (t_m / l))) * l;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.65d+80) then
        tmp = (((l / ((k * k) * t_m)) * l) / t_m) / t_m
    else
        tmp = (1.0d0 / (((k * t_m) * (k * t_m)) * (t_m / l))) * l
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.65e+80) {
		tmp = (((l / ((k * k) * t_m)) * l) / t_m) / t_m;
	} else {
		tmp = (1.0 / (((k * t_m) * (k * t_m)) * (t_m / l))) * l;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.65e+80:
		tmp = (((l / ((k * k) * t_m)) * l) / t_m) / t_m
	else:
		tmp = (1.0 / (((k * t_m) * (k * t_m)) * (t_m / l))) * l
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.65e+80)
		tmp = Float64(Float64(Float64(Float64(l / Float64(Float64(k * k) * t_m)) * l) / t_m) / t_m);
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(k * t_m) * Float64(k * t_m)) * Float64(t_m / 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 <= 1.65e+80)
		tmp = (((l / ((k * k) * t_m)) * l) / t_m) / t_m;
	else
		tmp = (1.0 / (((k * t_m) * (k * t_m)) * (t_m / l))) * l;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.65e+80], N[(N[(N[(N[(l / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision], N[(N[(1.0 / N[(N[(N[(k * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.64999999999999995e80

   1. Initial program 57.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right)} \cdot {t}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 53.1

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   5. Applied rewrites 53.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 61.5%

      \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t}}{t}}{\color{blue}{t}} \]

   if 1.64999999999999995e80 < t

   1. Initial program 67.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. 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. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right)} \cdot {t}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 58.9

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   5. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 86.7%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 86.7%

      \[\leadsto \frac{1}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot k\right) \cdot \frac{t}{\ell}} \cdot \ell \]
   10. Step-by-step derivation

   11. Applied rewrites 91.1%

       \[\leadsto \frac{1}{\left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right) \cdot \frac{t}{\ell}} \cdot \ell \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.65 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \ell}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \frac{t}{\ell}} \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 64.9% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\ell}{\left(t\_m \cdot t\_m\right) \cdot k} \cdot \frac{\ell}{k \cdot t\_m}\right) \end{array} \]

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 (* (* t_m t_m) k)) (/ l (* k t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / ((t_m * t_m) * k)) * (l / (k * 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 / ((t_m * t_m) * k)) * (l / (k * 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 / ((t_m * t_m) * k)) * (l / (k * 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 / ((t_m * t_m) * k)) * (l / (k * 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(l / Float64(Float64(t_m * t_m) * k)) * Float64(l / Float64(k * 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 / ((t_m * t_m) * k)) * (l / (k * 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[(l / N[(N[(t$95$m * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

   5. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)} \]

   6. associate-*r* N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right)} \cdot {t}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   12. lower-*.f64 54.1

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

5. Applied rewrites 54.1%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 69.0%

   \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]
8. Add Preprocessing

Alternative 17: 66.0% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\ell}{\left(\left(k \cdot t\_m\right) \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)} \cdot \ell\right) \end{array} \]

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 (* (* (* k t_m) t_m) (* k t_m))) l)))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / (((k * t_m) * t_m) * (k * t_m))) * l);
}

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 / (((k * t_m) * t_m) * (k * t_m))) * l)
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 / (((k * t_m) * t_m) * (k * t_m))) * l);
}

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 / (((k * t_m) * t_m) * (k * t_m))) * l)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l / Float64(Float64(Float64(k * t_m) * t_m) * Float64(k * t_m))) * l))
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 / (((k * t_m) * t_m) * (k * t_m))) * l);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l / N[(N[(N[(k * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

   5. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)} \]

   6. associate-*r* N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right)} \cdot {t}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   12. lower-*.f64 54.1

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

5. Applied rewrites 54.1%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 67.7%

   \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
8. Step-by-step derivation

9. Applied rewrites 68.8%

   \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
10. Add Preprocessing

Alternative 18: 57.7% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot \left(t\_m \cdot t\_m\right)} \cdot \ell\right) \end{array} \]

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 (* (* (* k k) t_m) (* t_m t_m))) l)))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / (((k * k) * t_m) * (t_m * t_m))) * l);
}

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 / (((k * k) * t_m) * (t_m * t_m))) * l)
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 / (((k * k) * t_m) * (t_m * t_m))) * l);
}

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 / (((k * k) * t_m) * (t_m * t_m))) * l)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l / Float64(Float64(Float64(k * k) * t_m) * Float64(t_m * t_m))) * l))
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 / (((k * k) * t_m) * (t_m * t_m))) * l);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

   5. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)} \]

   6. associate-*r* N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right)} \cdot {t}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   12. lower-*.f64 54.1

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

5. Applied rewrites 54.1%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 67.7%

   \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
8. Step-by-step derivation

9. Applied rewrites 58.0%

   \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(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))))