Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.1% → 84.9%

Time: 18.8s

Alternatives: 16

Speedup: 12.5×

• 26.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.05 \cdot 10^{-75}:\\ \;\;\;\;\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k \cdot \left(k \cdot \left(t\_m \cdot {\sin k}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \left(t\_m \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)}{\ell} \cdot \frac{t\_m \cdot \left(t\_m \cdot \sin k\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.05e-75)
    (/ (* (* 2.0 (* l l)) (cos k)) (* k (* k (* t_m (pow (sin k) 2.0)))))
    (/
     2.0
     (*
      (/ (* (tan k) (* t_m (fma k (/ k (* t_m t_m)) 2.0))) l)
      (/ (* t_m (* t_m (sin k))) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.05e-75) {
		tmp = ((2.0 * (l * l)) * cos(k)) / (k * (k * (t_m * pow(sin(k), 2.0))));
	} else {
		tmp = 2.0 / (((tan(k) * (t_m * fma(k, (k / (t_m * t_m)), 2.0))) / l) * ((t_m * (t_m * sin(k))) / l));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.05e-75)
		tmp = Float64(Float64(Float64(2.0 * Float64(l * l)) * cos(k)) / Float64(k * Float64(k * Float64(t_m * (sin(k) ^ 2.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(t_m * fma(k, Float64(k / Float64(t_m * t_m)), 2.0))) / l) * Float64(Float64(t_m * Float64(t_m * sin(k))) / l)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.05e-75], N[(N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t$95$m * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.0500000000000001e-75

   1. Initial program 49.2%

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

      1. associate-*l/ N/A

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

      2. div-inv N/A

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

      3. cube-mult N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 53.1

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

   4. Applied egg-rr 53.1%

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

      1. un-div-inv N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

      8. /-lowering-/.f64 66.1

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

   6. Applied egg-rr 66.1%

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

   7. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      15. sin-lowering-sin.f64 69.4

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

   9. Simplified 69.4%

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

   if 1.0500000000000001e-75 < t

   1. Initial program 69.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. associate-*r* N/A

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

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

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

   4. Applied egg-rr 65.1%

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

      1. un-div-inv N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

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

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

   6. Applied egg-rr 93.1%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-75}:\\ \;\;\;\;\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \left(t \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)}{\ell} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sin k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)\right)} \leq 5 \cdot 10^{+101}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot \ell}{\mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right) \cdot \left(\left(t\_m \cdot t\_2\right) \cdot \left(t\_m \cdot \tan k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t\_m \cdot \frac{t\_2 \cdot \frac{t\_m}{\ell}}{\ell}\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* t_m (sin k))))
   (*
    t_s
    (if (<=
         (/
          2.0
          (*
           (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0)
           (* (tan k) (* (sin k) (/ (pow t_m 3.0) (* l l))))))
         5e+101)
      (*
       l
       (/
        (* 2.0 l)
        (* (fma k (/ k (* t_m t_m)) 2.0) (* (* t_m t_2) (* t_m (tan k))))))
      (/ 2.0 (* 2.0 (* (tan k) (* t_m (/ (* t_2 (/ 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 t_2 = t_m * sin(k);
	double tmp;
	if ((2.0 / (((pow((k / t_m), 2.0) + 1.0) + 1.0) * (tan(k) * (sin(k) * (pow(t_m, 3.0) / (l * l)))))) <= 5e+101) {
		tmp = l * ((2.0 * l) / (fma(k, (k / (t_m * t_m)), 2.0) * ((t_m * t_2) * (t_m * tan(k)))));
	} else {
		tmp = 2.0 / (2.0 * (tan(k) * (t_m * ((t_2 * (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)
	t_2 = Float64(t_m * sin(k))
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0) * Float64(tan(k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))))) <= 5e+101)
		tmp = Float64(l * Float64(Float64(2.0 * l) / Float64(fma(k, Float64(k / Float64(t_m * t_m)), 2.0) * Float64(Float64(t_m * t_2) * Float64(t_m * tan(k))))));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(tan(k) * Float64(t_m * Float64(Float64(t_2 * Float64(t_m / l)) / l)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, 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[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+101], N[(l * N[(N[(2.0 * l), $MachinePrecision] / N[(N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(t$95$m * t$95$2), $MachinePrecision] * N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$2 * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)))) < 4.99999999999999989e101

   1. Initial program 81.5%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. log-lowering-log.f64 20.2

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

   4. Applied egg-rr 20.2%

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

   6. Applied egg-rr 79.8%

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

      1. associate-*r* N/A

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

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

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

   8. Applied egg-rr 87.8%

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

   if 4.99999999999999989e101 < (/.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 23.7%

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

      1. associate-*l/ N/A

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

      2. div-inv N/A

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

      3. cube-mult N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 29.8

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

   4. Applied egg-rr 29.8%

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

      1. un-div-inv N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

      8. /-lowering-/.f64 55.7

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

   6. Applied egg-rr 55.7%

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

      1. *-commutative N/A

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

      2. frac-2neg N/A

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

      3. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. neg-lowering-neg.f64 55.8

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

   8. Applied egg-rr 55.8%

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

   9. Taylor expanded in k around 0

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

   11. Simplified 58.8%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 5 \cdot 10^{+101}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot \ell}{\mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right) \cdot \left(\left(t \cdot \left(t \cdot \sin k\right)\right) \cdot \left(t \cdot \tan k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t \cdot \frac{\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.9 \cdot 10^{-74}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \left(t\_m \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)}{\ell} \cdot \frac{t\_m \cdot \left(t\_m \cdot \sin k\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.9e-74)
    (* (* 2.0 (* l l)) (/ (cos k) (* (pow (sin k) 2.0) (* t_m (* k k)))))
    (/
     2.0
     (*
      (/ (* (tan k) (* t_m (fma k (/ k (* t_m t_m)) 2.0))) l)
      (/ (* t_m (* t_m (sin k))) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.9e-74) {
		tmp = (2.0 * (l * l)) * (cos(k) / (pow(sin(k), 2.0) * (t_m * (k * k))));
	} else {
		tmp = 2.0 / (((tan(k) * (t_m * fma(k, (k / (t_m * t_m)), 2.0))) / l) * ((t_m * (t_m * sin(k))) / l));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.9e-74)
		tmp = Float64(Float64(2.0 * Float64(l * l)) * Float64(cos(k) / Float64((sin(k) ^ 2.0) * Float64(t_m * Float64(k * k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(t_m * fma(k, Float64(k / Float64(t_m * t_m)), 2.0))) / l) * Float64(Float64(t_m * Float64(t_m * sin(k))) / l)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.9e-74], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t$95$m * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.8999999999999998e-74

   1. Initial program 49.2%

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

   3. Taylor expanded in 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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. /-lowering-/.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. cos-lowering-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. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 63.6

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

   5. Simplified 63.6%

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

   if 1.8999999999999998e-74 < t

   1. Initial program 69.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. associate-*r* N/A

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

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

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

   4. Applied egg-rr 65.1%

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

      1. un-div-inv N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

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

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

   6. Applied egg-rr 93.1%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.9 \cdot 10^{-74}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \left(t \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)}{\ell} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sin k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.05 \cdot 10^{-279}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t\_m \cdot \left(\frac{t\_2}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t\_m \leq 1.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(k \cdot k\right)\right) \cdot \frac{t\_2}{\left(\ell \cdot \ell\right) \cdot \cos k}}\\ \mathbf{elif}\;t\_m \leq 2.25 \cdot 10^{+110}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right) \cdot \left(\sin k \cdot \frac{t\_m \cdot t\_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t\_m \cdot \frac{t\_2 \cdot \frac{t\_m}{\ell}}{\ell}\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* t_m (sin k))))
   (*
    t_s
    (if (<= t_m 1.05e-279)
      (/ 2.0 (* 2.0 (* (tan k) (* t_m (* (/ t_2 l) (/ t_m l))))))
      (if (<= t_m 1.2e-145)
        (/ 2.0 (* (* k (* k k)) (/ t_2 (* (* l l) (cos k)))))
        (if (<= t_m 2.25e+110)
          (/
           2.0
           (*
            (/ t_m l)
            (*
             (* (tan k) (fma k (/ k (* t_m t_m)) 2.0))
             (* (sin k) (/ (* t_m t_m) l)))))
          (/ 2.0 (* 2.0 (* (tan k) (* t_m (/ (* t_2 (/ 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 t_2 = t_m * sin(k);
	double tmp;
	if (t_m <= 1.05e-279) {
		tmp = 2.0 / (2.0 * (tan(k) * (t_m * ((t_2 / l) * (t_m / l)))));
	} else if (t_m <= 1.2e-145) {
		tmp = 2.0 / ((k * (k * k)) * (t_2 / ((l * l) * cos(k))));
	} else if (t_m <= 2.25e+110) {
		tmp = 2.0 / ((t_m / l) * ((tan(k) * fma(k, (k / (t_m * t_m)), 2.0)) * (sin(k) * ((t_m * t_m) / l))));
	} else {
		tmp = 2.0 / (2.0 * (tan(k) * (t_m * ((t_2 * (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)
	t_2 = Float64(t_m * sin(k))
	tmp = 0.0
	if (t_m <= 1.05e-279)
		tmp = Float64(2.0 / Float64(2.0 * Float64(tan(k) * Float64(t_m * Float64(Float64(t_2 / l) * Float64(t_m / l))))));
	elseif (t_m <= 1.2e-145)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(k * k)) * Float64(t_2 / Float64(Float64(l * l) * cos(k)))));
	elseif (t_m <= 2.25e+110)
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)) * Float64(sin(k) * Float64(Float64(t_m * t_m) / l)))));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(tan(k) * Float64(t_m * Float64(Float64(t_2 * Float64(t_m / l)) / l)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.05e-279], N[(2.0 / N[(2.0 * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$2 / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.2e-145], N[(2.0 / N[(N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / N[(N[(l * l), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.25e+110], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$2 * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 1.05000000000000003e-279

   1. Initial program 53.5%

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

      1. associate-*l/ N/A

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

      2. div-inv N/A

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

      3. cube-mult N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 56.6

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

   4. Applied egg-rr 56.6%

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

      1. un-div-inv N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

      8. /-lowering-/.f64 68.9

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

   6. Applied egg-rr 68.9%

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

   7. Taylor expanded in k around 0

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

   9. Simplified 69.8%

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

   if 1.05000000000000003e-279 < t < 1.20000000000000008e-145

   1. Initial program 34.5%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 34.5

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

   5. Simplified 34.5%

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

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

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

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

      3. cube-mult N/A

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

      4. unpow2 N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 69.9

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

   8. Simplified 69.9%

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

   if 1.20000000000000008e-145 < t < 2.2500000000000001e110

   1. Initial program 66.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. associate-*r* N/A

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

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

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

   4. Applied egg-rr 66.6%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

   6. Applied egg-rr 75.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. associate-*r/ N/A

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

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

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

   8. Applied egg-rr 85.2%

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

   if 2.2500000000000001e110 < t

   1. Initial program 62.9%

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

      1. associate-*l/ N/A

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

      2. div-inv N/A

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

      3. cube-mult N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 70.0

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

   4. Applied egg-rr 70.0%

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

      1. un-div-inv N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

      8. /-lowering-/.f64 92.2

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

   6. Applied egg-rr 92.2%

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

      1. *-commutative N/A

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

      2. frac-2neg N/A

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

      3. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. neg-lowering-neg.f64 92.2

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

   8. Applied egg-rr 92.2%

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

   9. Taylor expanded in k around 0

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

   11. Simplified 92.2%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-279}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(k \cdot k\right)\right) \cdot \frac{t \cdot \sin k}{\left(\ell \cdot \ell\right) \cdot \cos k}}\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+110}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \left(\sin k \cdot \frac{t \cdot t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t \cdot \frac{\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+193}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m \cdot k}{\ell}\right)\right)\right) \cdot \left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t\_m \cdot \frac{\left(t\_m \cdot \sin k\right) \cdot \frac{t\_m}{\ell}}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 2e+193)
    (/
     2.0
     (*
      (* (tan k) (* t_m (* (/ t_m l) (/ (* t_m k) l))))
      (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0)))
    (/ 2.0 (* 2.0 (* (tan k) (* t_m (/ (* (* t_m (sin k)) (/ t_m l)) l))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 2e+193) {
		tmp = 2.0 / ((tan(k) * (t_m * ((t_m / l) * ((t_m * k) / l)))) * ((pow((k / t_m), 2.0) + 1.0) + 1.0));
	} else {
		tmp = 2.0 / (2.0 * (tan(k) * (t_m * (((t_m * sin(k)) * (t_m / l)) / l))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 2d+193) then
        tmp = 2.0d0 / ((tan(k) * (t_m * ((t_m / l) * ((t_m * k) / l)))) * ((((k / t_m) ** 2.0d0) + 1.0d0) + 1.0d0))
    else
        tmp = 2.0d0 / (2.0d0 * (tan(k) * (t_m * (((t_m * sin(k)) * (t_m / l)) / l))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 2e+193) {
		tmp = 2.0 / ((Math.tan(k) * (t_m * ((t_m / l) * ((t_m * k) / l)))) * ((Math.pow((k / t_m), 2.0) + 1.0) + 1.0));
	} else {
		tmp = 2.0 / (2.0 * (Math.tan(k) * (t_m * (((t_m * Math.sin(k)) * (t_m / l)) / l))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 2e+193:
		tmp = 2.0 / ((math.tan(k) * (t_m * ((t_m / l) * ((t_m * k) / l)))) * ((math.pow((k / t_m), 2.0) + 1.0) + 1.0))
	else:
		tmp = 2.0 / (2.0 * (math.tan(k) * (t_m * (((t_m * math.sin(k)) * (t_m / l)) / l))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 2e+193)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(t_m * Float64(Float64(t_m / l) * Float64(Float64(t_m * k) / l)))) * Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(tan(k) * Float64(t_m * Float64(Float64(Float64(t_m * sin(k)) * 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 ((l * l) <= 2e+193)
		tmp = 2.0 / ((tan(k) * (t_m * ((t_m / l) * ((t_m * k) / l)))) * ((((k / t_m) ^ 2.0) + 1.0) + 1.0));
	else
		tmp = 2.0 / (2.0 * (tan(k) * (t_m * (((t_m * sin(k)) * (t_m / l)) / l))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 2e+193], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 2.00000000000000013e193

   1. Initial program 60.6%

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

      1. associate-*l/ N/A

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

      2. div-inv N/A

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

      3. cube-mult N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 65.4

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

   4. Applied egg-rr 65.4%

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

      1. un-div-inv N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

      8. /-lowering-/.f64 74.6

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

   6. Applied egg-rr 74.6%

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

   7. Taylor expanded in k around 0

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

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

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

      2. *-lowering-*.f64 74.4

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

   9. Simplified 74.4%

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

   if 2.00000000000000013e193 < (*.f64 l l)

   1. Initial program 44.6%

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

      1. associate-*l/ N/A

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

      2. div-inv N/A

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

      3. cube-mult N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 49.8

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

   4. Applied egg-rr 49.8%

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

      1. un-div-inv N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

      8. /-lowering-/.f64 72.7

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

   6. Applied egg-rr 72.7%

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

      1. *-commutative N/A

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

      2. frac-2neg N/A

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

      3. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. neg-lowering-neg.f64 72.7

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

   8. Applied egg-rr 72.7%

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

   9. Taylor expanded in k around 0

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

   11. Simplified 84.8%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+193}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot k}{\ell}\right)\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t \cdot \frac{\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot \sin k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{t\_m \cdot \frac{\left(t\_m \cdot t\_2\right) \cdot \left(2 \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \left(t\_2 \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (* t_m (sin k)) l)))
   (*
    t_s
    (if (<= k 3.4e-24)
      (/ 2.0 (* t_m (/ (* (* t_m t_2) (* 2.0 k)) l)))
      (/
       2.0
       (*
        t_m
        (* (/ t_m l) (* t_2 (* (tan k) (fma k (/ k (* t_m t_m)) 2.0))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m * sin(k)) / l;
	double tmp;
	if (k <= 3.4e-24) {
		tmp = 2.0 / (t_m * (((t_m * t_2) * (2.0 * k)) / l));
	} else {
		tmp = 2.0 / (t_m * ((t_m / l) * (t_2 * (tan(k) * fma(k, (k / (t_m * t_m)), 2.0)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(t_m * sin(k)) / l)
	tmp = 0.0
	if (k <= 3.4e-24)
		tmp = Float64(2.0 / Float64(t_m * Float64(Float64(Float64(t_m * t_2) * Float64(2.0 * k)) / l)));
	else
		tmp = Float64(2.0 / Float64(t_m * Float64(Float64(t_m / l) * Float64(t_2 * Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 3.4e-24], N[(2.0 / N[(t$95$m * N[(N[(N[(t$95$m * t$95$2), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$2 * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.39999999999999992e-24

   1. Initial program 57.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. associate-*r* N/A

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

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

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

   4. Applied egg-rr 47.4%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

   6. Applied egg-rr 58.4%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 67.8

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

   9. Simplified 67.8%

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

       1. *-commutative N/A

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

       2. associate-/r* N/A

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

       3. associate-*l/ N/A

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

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

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

   11. Applied egg-rr 77.3%

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

   if 3.39999999999999992e-24 < k

   1. Initial program 50.5%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. associate-*r* N/A

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

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

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

   4. Applied egg-rr 45.3%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

   6. Applied egg-rr 53.4%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. metadata-eval N/A

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

      11. associate-+l+ N/A

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

      12. associate-*r/ N/A

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

      13. frac-times N/A

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

      14. unpow2 N/A

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

      15. +-commutative N/A

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

   8. Applied egg-rr 74.4%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{t \cdot \frac{\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \left(2 \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{t\_m \cdot \frac{\left(t\_m \cdot \frac{t\_m \cdot \sin k}{\ell}\right) \cdot \left(2 \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left(\left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right) \cdot \left(\left(t\_m \cdot \frac{t\_m}{\ell}\right) \cdot \frac{\sin k}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.4e-24)
    (/ 2.0 (* t_m (/ (* (* t_m (/ (* t_m (sin k)) l)) (* 2.0 k)) l)))
    (/
     2.0
     (*
      t_m
      (*
       (* (tan k) (fma k (/ k (* t_m t_m)) 2.0))
       (* (* t_m (/ t_m l)) (/ (sin k) l))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.4e-24) {
		tmp = 2.0 / (t_m * (((t_m * ((t_m * sin(k)) / l)) * (2.0 * k)) / l));
	} else {
		tmp = 2.0 / (t_m * ((tan(k) * fma(k, (k / (t_m * t_m)), 2.0)) * ((t_m * (t_m / l)) * (sin(k) / l))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3.4e-24)
		tmp = Float64(2.0 / Float64(t_m * Float64(Float64(Float64(t_m * Float64(Float64(t_m * sin(k)) / l)) * Float64(2.0 * k)) / l)));
	else
		tmp = Float64(2.0 / Float64(t_m * Float64(Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)) * Float64(Float64(t_m * Float64(t_m / l)) * Float64(sin(k) / l)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 3.4e-24], N[(2.0 / N[(t$95$m * N[(N[(N[(t$95$m * N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$m * N[(N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.39999999999999992e-24

   1. Initial program 57.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. associate-*r* N/A

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

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

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

   4. Applied egg-rr 47.4%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

   6. Applied egg-rr 58.4%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 67.8

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

   9. Simplified 67.8%

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

       1. *-commutative N/A

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

       2. associate-/r* N/A

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

       3. associate-*l/ N/A

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

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

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

   11. Applied egg-rr 77.3%

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

   if 3.39999999999999992e-24 < k

   1. Initial program 50.5%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. associate-*r* N/A

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

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

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

   4. Applied egg-rr 45.3%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

   6. Applied egg-rr 53.4%

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

      1. times-frac N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. sin-lowering-sin.f64 71.4

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

   8. Applied egg-rr 71.4%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{t \cdot \frac{\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \left(2 \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(\left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \frac{\sin k}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sin k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.3 \cdot 10^{-280}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t\_m \cdot \left(\frac{t\_2}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t\_m \leq 3.1 \cdot 10^{-74}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(k \cdot k\right)\right) \cdot \frac{t\_2}{\left(\ell \cdot \ell\right) \cdot \cos k}}\\ \mathbf{elif}\;t\_m \leq 1.46 \cdot 10^{+75}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\ell}{k \cdot \left(t\_m \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t\_m \cdot \frac{t\_2 \cdot \frac{t\_m}{\ell}}{\ell}\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* t_m (sin k))))
   (*
    t_s
    (if (<= t_m 7.3e-280)
      (/ 2.0 (* 2.0 (* (tan k) (* t_m (* (/ t_2 l) (/ t_m l))))))
      (if (<= t_m 3.1e-74)
        (/ 2.0 (* (* k (* k k)) (/ t_2 (* (* l l) (cos k)))))
        (if (<= t_m 1.46e+75)
          (* (/ l (* t_m k)) (/ l (* k (* t_m t_m))))
          (/ 2.0 (* 2.0 (* (tan k) (* t_m (/ (* t_2 (/ 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 t_2 = t_m * sin(k);
	double tmp;
	if (t_m <= 7.3e-280) {
		tmp = 2.0 / (2.0 * (tan(k) * (t_m * ((t_2 / l) * (t_m / l)))));
	} else if (t_m <= 3.1e-74) {
		tmp = 2.0 / ((k * (k * k)) * (t_2 / ((l * l) * cos(k))));
	} else if (t_m <= 1.46e+75) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else {
		tmp = 2.0 / (2.0 * (tan(k) * (t_m * ((t_2 * (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) :: t_2
    real(8) :: tmp
    t_2 = t_m * sin(k)
    if (t_m <= 7.3d-280) then
        tmp = 2.0d0 / (2.0d0 * (tan(k) * (t_m * ((t_2 / l) * (t_m / l)))))
    else if (t_m <= 3.1d-74) then
        tmp = 2.0d0 / ((k * (k * k)) * (t_2 / ((l * l) * cos(k))))
    else if (t_m <= 1.46d+75) then
        tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)))
    else
        tmp = 2.0d0 / (2.0d0 * (tan(k) * (t_m * ((t_2 * (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 t_2 = t_m * Math.sin(k);
	double tmp;
	if (t_m <= 7.3e-280) {
		tmp = 2.0 / (2.0 * (Math.tan(k) * (t_m * ((t_2 / l) * (t_m / l)))));
	} else if (t_m <= 3.1e-74) {
		tmp = 2.0 / ((k * (k * k)) * (t_2 / ((l * l) * Math.cos(k))));
	} else if (t_m <= 1.46e+75) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else {
		tmp = 2.0 / (2.0 * (Math.tan(k) * (t_m * ((t_2 * (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):
	t_2 = t_m * math.sin(k)
	tmp = 0
	if t_m <= 7.3e-280:
		tmp = 2.0 / (2.0 * (math.tan(k) * (t_m * ((t_2 / l) * (t_m / l)))))
	elif t_m <= 3.1e-74:
		tmp = 2.0 / ((k * (k * k)) * (t_2 / ((l * l) * math.cos(k))))
	elif t_m <= 1.46e+75:
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)))
	else:
		tmp = 2.0 / (2.0 * (math.tan(k) * (t_m * ((t_2 * (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)
	t_2 = Float64(t_m * sin(k))
	tmp = 0.0
	if (t_m <= 7.3e-280)
		tmp = Float64(2.0 / Float64(2.0 * Float64(tan(k) * Float64(t_m * Float64(Float64(t_2 / l) * Float64(t_m / l))))));
	elseif (t_m <= 3.1e-74)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(k * k)) * Float64(t_2 / Float64(Float64(l * l) * cos(k)))));
	elseif (t_m <= 1.46e+75)
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(l / Float64(k * Float64(t_m * t_m))));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(tan(k) * Float64(t_m * Float64(Float64(t_2 * 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)
	t_2 = t_m * sin(k);
	tmp = 0.0;
	if (t_m <= 7.3e-280)
		tmp = 2.0 / (2.0 * (tan(k) * (t_m * ((t_2 / l) * (t_m / l)))));
	elseif (t_m <= 3.1e-74)
		tmp = 2.0 / ((k * (k * k)) * (t_2 / ((l * l) * cos(k))));
	elseif (t_m <= 1.46e+75)
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	else
		tmp = 2.0 / (2.0 * (tan(k) * (t_m * ((t_2 * (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_] := Block[{t$95$2 = N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.3e-280], N[(2.0 / N[(2.0 * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$2 / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.1e-74], N[(2.0 / N[(N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / N[(N[(l * l), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.46e+75], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$2 * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 7.30000000000000023e-280

   1. Initial program 53.5%

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

      1. associate-*l/ N/A

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

      2. div-inv N/A

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

      3. cube-mult N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 56.6

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

   4. Applied egg-rr 56.6%

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

      1. un-div-inv N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

      8. /-lowering-/.f64 68.9

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

   6. Applied egg-rr 68.9%

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

   7. Taylor expanded in k around 0

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

   9. Simplified 69.8%

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

   if 7.30000000000000023e-280 < t < 3.1000000000000002e-74

   1. Initial program 37.7%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 35.7

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

   5. Simplified 35.7%

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

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

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

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

      3. cube-mult N/A

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

      4. unpow2 N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 65.6

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

   8. Simplified 65.6%

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

   if 3.1000000000000002e-74 < t < 1.4600000000000001e75

   1. Initial program 80.3%

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

   3. Taylor expanded in k around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 68.9

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

   5. Simplified 68.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. associate-*l* N/A

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

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

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

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

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

      11. *-lowering-*.f64 77.7

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

   7. Applied egg-rr 77.7%

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

      1. associate-*l/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. *-lowering-*.f64 88.9

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

   9. Applied egg-rr 88.9%

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

   if 1.4600000000000001e75 < t

   1. Initial program 61.2%

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

      1. associate-*l/ N/A

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

      2. div-inv N/A

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

      3. cube-mult N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 69.8

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

   4. Applied egg-rr 69.8%

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

      1. un-div-inv N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

      8. /-lowering-/.f64 92.7

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

   6. Applied egg-rr 92.7%

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

      1. *-commutative N/A

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

      2. frac-2neg N/A

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

      3. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. neg-lowering-neg.f64 92.7

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

   8. Applied egg-rr 92.7%

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

   9. Taylor expanded in k around 0

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

   11. Simplified 90.7%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.3 \cdot 10^{-280}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-74}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(k \cdot k\right)\right) \cdot \frac{t \cdot \sin k}{\left(\ell \cdot \ell\right) \cdot \cos k}}\\ \mathbf{elif}\;t \leq 1.46 \cdot 10^{+75}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\ell}{k \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t \cdot \frac{\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sin k\\ t_3 := \frac{2}{2 \cdot \left(\tan k \cdot \left(t\_m \cdot \left(\frac{t\_2}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)\right)}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.5 \cdot 10^{-279}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_m \leq 6.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(k \cdot k\right)\right) \cdot \frac{t\_2}{\left(\ell \cdot \ell\right) \cdot \cos k}}\\ \mathbf{elif}\;t\_m \leq 4.6 \cdot 10^{+75}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\ell}{k \cdot \left(t\_m \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* t_m (sin k)))
        (t_3 (/ 2.0 (* 2.0 (* (tan k) (* t_m (* (/ t_2 l) (/ t_m l))))))))
   (*
    t_s
    (if (<= t_m 2.5e-279)
      t_3
      (if (<= t_m 6.2e-72)
        (/ 2.0 (* (* k (* k k)) (/ t_2 (* (* l l) (cos k)))))
        (if (<= t_m 4.6e+75)
          (* (/ l (* t_m k)) (/ l (* k (* t_m t_m))))
          t_3))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * sin(k);
	double t_3 = 2.0 / (2.0 * (tan(k) * (t_m * ((t_2 / l) * (t_m / l)))));
	double tmp;
	if (t_m <= 2.5e-279) {
		tmp = t_3;
	} else if (t_m <= 6.2e-72) {
		tmp = 2.0 / ((k * (k * k)) * (t_2 / ((l * l) * cos(k))));
	} else if (t_m <= 4.6e+75) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else {
		tmp = t_3;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = t_m * sin(k)
    t_3 = 2.0d0 / (2.0d0 * (tan(k) * (t_m * ((t_2 / l) * (t_m / l)))))
    if (t_m <= 2.5d-279) then
        tmp = t_3
    else if (t_m <= 6.2d-72) then
        tmp = 2.0d0 / ((k * (k * k)) * (t_2 / ((l * l) * cos(k))))
    else if (t_m <= 4.6d+75) then
        tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)))
    else
        tmp = t_3
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * Math.sin(k);
	double t_3 = 2.0 / (2.0 * (Math.tan(k) * (t_m * ((t_2 / l) * (t_m / l)))));
	double tmp;
	if (t_m <= 2.5e-279) {
		tmp = t_3;
	} else if (t_m <= 6.2e-72) {
		tmp = 2.0 / ((k * (k * k)) * (t_2 / ((l * l) * Math.cos(k))));
	} else if (t_m <= 4.6e+75) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else {
		tmp = t_3;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = t_m * math.sin(k)
	t_3 = 2.0 / (2.0 * (math.tan(k) * (t_m * ((t_2 / l) * (t_m / l)))))
	tmp = 0
	if t_m <= 2.5e-279:
		tmp = t_3
	elif t_m <= 6.2e-72:
		tmp = 2.0 / ((k * (k * k)) * (t_2 / ((l * l) * math.cos(k))))
	elif t_m <= 4.6e+75:
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)))
	else:
		tmp = t_3
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m * sin(k))
	t_3 = Float64(2.0 / Float64(2.0 * Float64(tan(k) * Float64(t_m * Float64(Float64(t_2 / l) * Float64(t_m / l))))))
	tmp = 0.0
	if (t_m <= 2.5e-279)
		tmp = t_3;
	elseif (t_m <= 6.2e-72)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(k * k)) * Float64(t_2 / Float64(Float64(l * l) * cos(k)))));
	elseif (t_m <= 4.6e+75)
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(l / Float64(k * Float64(t_m * t_m))));
	else
		tmp = t_3;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = t_m * sin(k);
	t_3 = 2.0 / (2.0 * (tan(k) * (t_m * ((t_2 / l) * (t_m / l)))));
	tmp = 0.0;
	if (t_m <= 2.5e-279)
		tmp = t_3;
	elseif (t_m <= 6.2e-72)
		tmp = 2.0 / ((k * (k * k)) * (t_2 / ((l * l) * cos(k))));
	elseif (t_m <= 4.6e+75)
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	else
		tmp = t_3;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 / N[(2.0 * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$2 / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.5e-279], t$95$3, If[LessEqual[t$95$m, 6.2e-72], N[(2.0 / N[(N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / N[(N[(l * l), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.6e+75], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.49999999999999984e-279 or 4.5999999999999997e75 < t

   1. Initial program 55.6%

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

      1. associate-*l/ N/A

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

      2. div-inv N/A

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

      3. cube-mult N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 60.2

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

   4. Applied egg-rr 60.2%

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

      1. un-div-inv N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

      8. /-lowering-/.f64 75.4

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

   6. Applied egg-rr 75.4%

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

   7. Taylor expanded in k around 0

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

   9. Simplified 75.4%

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

   if 2.49999999999999984e-279 < t < 6.1999999999999996e-72

   1. Initial program 37.7%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \frac{k}{{\ell}^{2}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\frac{k}{{\ell}^{2}}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\color{blue}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. *-lowering-*.f64 35.7

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\color{blue}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Simplified 35.7%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2} \cdot \cos k}}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{3} \cdot \frac{t \cdot \sin k}{{\ell}^{2} \cdot \cos k}}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{3} \cdot \frac{t \cdot \sin k}{{\ell}^{2} \cdot \cos k}}} \]

      3. cube-mult N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \left(k \cdot k\right)\right)} \cdot \frac{t \cdot \sin k}{{\ell}^{2} \cdot \cos k}} \]

      4. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot \color{blue}{{k}^{2}}\right) \cdot \frac{t \cdot \sin k}{{\ell}^{2} \cdot \cos k}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot {k}^{2}\right)} \cdot \frac{t \cdot \sin k}{{\ell}^{2} \cdot \cos k}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{t \cdot \sin k}{{\ell}^{2} \cdot \cos k}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{t \cdot \sin k}{{\ell}^{2} \cdot \cos k}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{t \cdot \sin k}{{\ell}^{2} \cdot \cos k}}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{t \cdot \sin k}}{{\ell}^{2} \cdot \cos k}} \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot k\right)\right) \cdot \frac{t \cdot \color{blue}{\sin k}}{{\ell}^{2} \cdot \cos k}} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot k\right)\right) \cdot \frac{t \cdot \sin k}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot k\right)\right) \cdot \frac{t \cdot \sin k}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot k\right)\right) \cdot \frac{t \cdot \sin k}{\color{blue}{\cos k} \cdot {\ell}^{2}}} \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot k\right)\right) \cdot \frac{t \cdot \sin k}{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \]

      15. *-lowering-*.f64 65.6

          \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot k\right)\right) \cdot \frac{t \cdot \sin k}{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \]

   8. Simplified 65.6%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \left(k \cdot k\right)\right) \cdot \frac{t \cdot \sin k}{\cos k \cdot \left(\ell \cdot \ell\right)}}} \]

   if 6.1999999999999996e-72 < t < 4.5999999999999997e75

   1. Initial program 80.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. *-lowering-*.f64 68.9

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 68.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot k}} \cdot \ell \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right)} \cdot \ell \]

      11. *-lowering-*.f64 77.7

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot k\right)\right)} \cdot \ell \]

   7. Applied egg-rr 77.7%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot k\right)}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot t}} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot t}} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot k}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot t} \cdot \frac{\ell}{\color{blue}{k \cdot \left(t \cdot t\right)}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot t} \cdot \frac{\ell}{\color{blue}{k \cdot \left(t \cdot t\right)}} \]

      10. *-lowering-*.f64 88.9

          \[\leadsto \frac{\ell}{k \cdot t} \cdot \frac{\ell}{k \cdot \color{blue}{\left(t \cdot t\right)}} \]

   9. Applied egg-rr 88.9%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{k \cdot \left(t \cdot t\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.5 \cdot 10^{-279}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(k \cdot k\right)\right) \cdot \frac{t \cdot \sin k}{\left(\ell \cdot \ell\right) \cdot \cos k}}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+75}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\ell}{k \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{+56}:\\ \;\;\;\;\frac{2}{t\_m \cdot \frac{\left(t\_m \cdot \frac{t\_m \cdot \sin k}{\ell}\right) \cdot \left(2 \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{k \cdot \left(t\_m \cdot k\right)}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.45e+56)
    (/ 2.0 (* t_m (/ (* (* t_m (/ (* t_m (sin k)) l)) (* 2.0 k)) l)))
    (/ 2.0 (* (* k k) (/ (* k (* t_m k)) (* l l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.45e+56) {
		tmp = 2.0 / (t_m * (((t_m * ((t_m * sin(k)) / l)) * (2.0 * k)) / l));
	} else {
		tmp = 2.0 / ((k * k) * ((k * (t_m * k)) / (l * l)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.45d+56) then
        tmp = 2.0d0 / (t_m * (((t_m * ((t_m * sin(k)) / l)) * (2.0d0 * k)) / l))
    else
        tmp = 2.0d0 / ((k * k) * ((k * (t_m * k)) / (l * l)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.45e+56) {
		tmp = 2.0 / (t_m * (((t_m * ((t_m * Math.sin(k)) / l)) * (2.0 * k)) / l));
	} else {
		tmp = 2.0 / ((k * k) * ((k * (t_m * k)) / (l * l)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.45e+56:
		tmp = 2.0 / (t_m * (((t_m * ((t_m * math.sin(k)) / l)) * (2.0 * k)) / l))
	else:
		tmp = 2.0 / ((k * k) * ((k * (t_m * k)) / (l * l)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.45e+56)
		tmp = Float64(2.0 / Float64(t_m * Float64(Float64(Float64(t_m * Float64(Float64(t_m * sin(k)) / l)) * Float64(2.0 * k)) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(k * Float64(t_m * k)) / Float64(l * l))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.45e+56)
		tmp = 2.0 / (t_m * (((t_m * ((t_m * sin(k)) / l)) * (2.0 * k)) / l));
	else
		tmp = 2.0 / ((k * k) * ((k * (t_m * k)) / (l * l)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.45e+56], N[(2.0 / N[(t$95$m * N[(N[(N[(t$95$m * N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.45000000000000004e56

   1. Initial program 57.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}} \]

      4. div-inv N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \frac{1}{\ell \cdot \ell}\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \left({t}^{3} \cdot \sin k\right)\right) \cdot \frac{1}{\ell \cdot \ell}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \left({t}^{3} \cdot \sin k\right)\right) \cdot \frac{1}{\ell \cdot \ell}}} \]

   4. Applied egg-rr 48.9%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right) \cdot \left(t \cdot \left(t \cdot \left(t \cdot \sin k\right)\right)\right)\right) \cdot \frac{1}{\ell \cdot \ell}}} \]
   5. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right) \cdot \left(\left(t \cdot \left(t \cdot \left(t \cdot \sin k\right)\right)\right) \cdot \frac{1}{\ell \cdot \ell}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right) \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot \left(t \cdot \sin k\right)\right) \cdot \frac{1}{\ell \cdot \ell}\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right) \cdot \color{blue}{\left(\left(\left(t \cdot \left(t \cdot \sin k\right)\right) \cdot \frac{1}{\ell \cdot \ell}\right) \cdot t\right)}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right) \cdot \left(\left(t \cdot \left(t \cdot \sin k\right)\right) \cdot \frac{1}{\ell \cdot \ell}\right)\right) \cdot t}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right) \cdot \left(\left(t \cdot \left(t \cdot \sin k\right)\right) \cdot \frac{1}{\ell \cdot \ell}\right)\right) \cdot t}} \]

   6. Applied egg-rr 58.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell \cdot \ell}\right) \cdot t}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\color{blue}{\left(2 \cdot k\right)} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell \cdot \ell}\right) \cdot t} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot 2\right)} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell \cdot \ell}\right) \cdot t} \]

      2. *-lowering-*.f64 66.4

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot 2\right)} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell \cdot \ell}\right) \cdot t} \]

   9. Simplified 66.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot 2\right)} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell \cdot \ell}\right) \cdot t} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell \cdot \ell} \cdot \left(k \cdot 2\right)\right)} \cdot t} \]

       2. associate-/r* N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{t \cdot \left(t \cdot \sin k\right)}{\ell}}{\ell}} \cdot \left(k \cdot 2\right)\right) \cdot t} \]

       3. associate-*l/ N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{t \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(k \cdot 2\right)}{\ell}} \cdot t} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{t \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(k \cdot 2\right)}{\ell}} \cdot t} \]

   11. Applied egg-rr 75.4%

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \left(k \cdot 2\right)}{\ell}} \cdot t} \]

   if 1.45000000000000004e56 < k

   1. Initial program 46.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\mathsf{fma}\left(2, \frac{{t}^{3}}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \color{blue}{\frac{{t}^{3}}{{\ell}^{2}}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      6. cube-mult N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \color{blue}{{t}^{2}}}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{\color{blue}{t \cdot {t}^{2}}}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \color{blue}{\left(t \cdot t\right)}}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \color{blue}{\left(t \cdot t\right)}}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell}, \color{blue}{\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}}\right)} \]

   5. Simplified 52.8%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell}, \frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t, \left(t \cdot t\right) \cdot 0.3333333333333333, t\right)}{\ell \cdot \ell}\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{{\ell}^{2}}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{{\ell}^{2}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{{\ell}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{k \cdot \color{blue}{\left(k \cdot t\right)}}{{\ell}^{2}}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{k \cdot \left(k \cdot t\right)}{\color{blue}{\ell \cdot \ell}}} \]

      7. *-lowering-*.f64 62.3

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{k \cdot \left(k \cdot t\right)}{\color{blue}{\ell \cdot \ell}}} \]

   8. Simplified 62.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{+56}:\\ \;\;\;\;\frac{2}{t \cdot \frac{\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \left(2 \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{k \cdot \left(t \cdot k\right)}{\ell \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.6% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.7 \cdot 10^{-71}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{k \cdot \left(t\_m \cdot k\right)}{\ell \cdot \ell}}\\ \mathbf{elif}\;t\_m \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\ell}{k \cdot \left(t\_m \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.7e-71)
    (/ 2.0 (* (* k k) (/ (* k (* t_m k)) (* l l))))
    (if (<= t_m 5e+132)
      (* (/ l (* t_m k)) (/ l (* k (* t_m t_m))))
      (* l (/ l (* t_m (* (* t_m k) (* t_m k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.7e-71) {
		tmp = 2.0 / ((k * k) * ((k * (t_m * k)) / (l * l)));
	} else if (t_m <= 5e+132) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else {
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.7d-71) then
        tmp = 2.0d0 / ((k * k) * ((k * (t_m * k)) / (l * l)))
    else if (t_m <= 5d+132) then
        tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)))
    else
        tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.7e-71) {
		tmp = 2.0 / ((k * k) * ((k * (t_m * k)) / (l * l)));
	} else if (t_m <= 5e+132) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else {
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.7e-71:
		tmp = 2.0 / ((k * k) * ((k * (t_m * k)) / (l * l)))
	elif t_m <= 5e+132:
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)))
	else:
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.7e-71)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(k * Float64(t_m * k)) / Float64(l * l))));
	elseif (t_m <= 5e+132)
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(l / Float64(k * Float64(t_m * t_m))));
	else
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(Float64(t_m * k) * Float64(t_m * k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.7e-71)
		tmp = 2.0 / ((k * k) * ((k * (t_m * k)) / (l * l)));
	elseif (t_m <= 5e+132)
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	else
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.7e-71], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5e+132], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(t$95$m * N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.70000000000000002e-71

   1. Initial program 49.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\mathsf{fma}\left(2, \frac{{t}^{3}}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \color{blue}{\frac{{t}^{3}}{{\ell}^{2}}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      6. cube-mult N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \color{blue}{{t}^{2}}}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{\color{blue}{t \cdot {t}^{2}}}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \color{blue}{\left(t \cdot t\right)}}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \color{blue}{\left(t \cdot t\right)}}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell}, \color{blue}{\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}}\right)} \]

   5. Simplified 55.8%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell}, \frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t, \left(t \cdot t\right) \cdot 0.3333333333333333, t\right)}{\ell \cdot \ell}\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{{\ell}^{2}}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{{\ell}^{2}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{{\ell}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{k \cdot \color{blue}{\left(k \cdot t\right)}}{{\ell}^{2}}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{k \cdot \left(k \cdot t\right)}{\color{blue}{\ell \cdot \ell}}} \]

      7. *-lowering-*.f64 56.8

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{k \cdot \left(k \cdot t\right)}{\color{blue}{\ell \cdot \ell}}} \]

   8. Simplified 56.8%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell}}} \]

   if 1.70000000000000002e-71 < t < 5.0000000000000001e132

   1. Initial program 75.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. *-lowering-*.f64 66.2

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 66.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot k}} \cdot \ell \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right)} \cdot \ell \]

      11. *-lowering-*.f64 77.8

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot k\right)\right)} \cdot \ell \]

   7. Applied egg-rr 77.8%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot k\right)}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot t}} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot t}} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot k}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot t} \cdot \frac{\ell}{\color{blue}{k \cdot \left(t \cdot t\right)}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot t} \cdot \frac{\ell}{\color{blue}{k \cdot \left(t \cdot t\right)}} \]

      10. *-lowering-*.f64 86.6

          \[\leadsto \frac{\ell}{k \cdot t} \cdot \frac{\ell}{k \cdot \color{blue}{\left(t \cdot t\right)}} \]

   9. Applied egg-rr 86.6%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{k \cdot \left(t \cdot t\right)}} \]

   if 5.0000000000000001e132 < t

   1. Initial program 62.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. *-lowering-*.f64 56.3

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 56.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot k}} \cdot \ell \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right)} \cdot \ell \]

      11. *-lowering-*.f64 70.7

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot k\right)\right)} \cdot \ell \]

   7. Applied egg-rr 70.7%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}\right)} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]

      4. *-lowering-*.f64 77.9

         \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot t\right)\right)} \cdot \ell \]

   9. Applied egg-rr 77.9%

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(\left(k \cdot t\right) \cdot t\right)}} \cdot \ell \]

       2. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t}} \cdot \ell \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t}} \cdot \ell \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot t} \cdot \ell \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot t\right)\right) \cdot t} \cdot \ell \]

       6. *-lowering-*.f64 88.1

          \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot t} \cdot \ell \]

   11. Applied egg-rr 88.1%

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t}} \cdot \ell \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{-71}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{k \cdot \left(t \cdot k\right)}{\ell \cdot \ell}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\ell}{k \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.1% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t\_m}}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;t\_m \leq 8.5 \cdot 10^{+133}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\ell}{k \cdot \left(t\_m \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5e-69)
    (/ (/ (* l l) t_m) (* t_m (* t_m (* k k))))
    (if (<= t_m 8.5e+133)
      (* (/ l (* t_m k)) (/ l (* k (* t_m t_m))))
      (* l (/ l (* t_m (* (* t_m k) (* t_m k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5e-69) {
		tmp = ((l * l) / t_m) / (t_m * (t_m * (k * k)));
	} else if (t_m <= 8.5e+133) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else {
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 5d-69) then
        tmp = ((l * l) / t_m) / (t_m * (t_m * (k * k)))
    else if (t_m <= 8.5d+133) then
        tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)))
    else
        tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5e-69) {
		tmp = ((l * l) / t_m) / (t_m * (t_m * (k * k)));
	} else if (t_m <= 8.5e+133) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else {
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 5e-69:
		tmp = ((l * l) / t_m) / (t_m * (t_m * (k * k)))
	elif t_m <= 8.5e+133:
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)))
	else:
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5e-69)
		tmp = Float64(Float64(Float64(l * l) / t_m) / Float64(t_m * Float64(t_m * Float64(k * k))));
	elseif (t_m <= 8.5e+133)
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(l / Float64(k * Float64(t_m * t_m))));
	else
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(Float64(t_m * k) * Float64(t_m * k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 5e-69)
		tmp = ((l * l) / t_m) / (t_m * (t_m * (k * k)));
	elseif (t_m <= 8.5e+133)
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	else
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5e-69], N[(N[(N[(l * l), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8.5e+133], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(t$95$m * N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.00000000000000033e-69

   1. Initial program 49.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. *-lowering-*.f64 49.8

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 49.8%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\ell \cdot \ell}{t}}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{\color{blue}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{\color{blue}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \]

      9. *-lowering-*.f64 59.5

         \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{t \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   7. Applied egg-rr 59.5%

      \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

   if 5.00000000000000033e-69 < t < 8.50000000000000044e133

   1. Initial program 75.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. *-lowering-*.f64 66.2

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 66.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot k}} \cdot \ell \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right)} \cdot \ell \]

      11. *-lowering-*.f64 77.8

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot k\right)\right)} \cdot \ell \]

   7. Applied egg-rr 77.8%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot k\right)}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot t}} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot t}} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot k}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot t} \cdot \frac{\ell}{\color{blue}{k \cdot \left(t \cdot t\right)}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot t} \cdot \frac{\ell}{\color{blue}{k \cdot \left(t \cdot t\right)}} \]

      10. *-lowering-*.f64 86.6

          \[\leadsto \frac{\ell}{k \cdot t} \cdot \frac{\ell}{k \cdot \color{blue}{\left(t \cdot t\right)}} \]

   9. Applied egg-rr 86.6%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{k \cdot \left(t \cdot t\right)}} \]

   if 8.50000000000000044e133 < t

   1. Initial program 62.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. *-lowering-*.f64 56.3

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 56.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot k}} \cdot \ell \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right)} \cdot \ell \]

      11. *-lowering-*.f64 70.7

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot k\right)\right)} \cdot \ell \]

   7. Applied egg-rr 70.7%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}\right)} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]

      4. *-lowering-*.f64 77.9

         \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot t\right)\right)} \cdot \ell \]

   9. Applied egg-rr 77.9%

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(\left(k \cdot t\right) \cdot t\right)}} \cdot \ell \]

       2. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t}} \cdot \ell \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t}} \cdot \ell \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot t} \cdot \ell \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot t\right)\right) \cdot t} \cdot \ell \]

       6. *-lowering-*.f64 88.1

          \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot t} \cdot \ell \]

   11. Applied egg-rr 88.1%

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t}} \cdot \ell \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t}}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+133}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\ell}{k \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.1% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.26 \cdot 10^{-156}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.26e-156)
    (* l (/ l (* t_m (* (* t_m k) (* t_m k)))))
    (* (/ l t_m) (/ l (* t_m (* t_m (* k k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.26e-156) {
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	} else {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.26d-156) then
        tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))))
    else
        tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.26e-156) {
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	} else {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 2.26e-156:
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))))
	else:
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.26e-156)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(Float64(t_m * k) * Float64(t_m * k)))));
	else
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(t_m * Float64(t_m * Float64(k * k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 2.26e-156)
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	else
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.26e-156], N[(l * N[(l / N[(t$95$m * N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.26e-156

   1. Initial program 59.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.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. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. *-lowering-*.f64 54.0

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 54.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot k}} \cdot \ell \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right)} \cdot \ell \]

      11. *-lowering-*.f64 68.4

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot k\right)\right)} \cdot \ell \]

   7. Applied egg-rr 68.4%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}\right)} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]

      4. *-lowering-*.f64 71.4

         \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot t\right)\right)} \cdot \ell \]

   9. Applied egg-rr 71.4%

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(\left(k \cdot t\right) \cdot t\right)}} \cdot \ell \]

       2. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t}} \cdot \ell \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t}} \cdot \ell \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot t} \cdot \ell \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot t\right)\right) \cdot t} \cdot \ell \]

       6. *-lowering-*.f64 74.4

          \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot t} \cdot \ell \]

   11. Applied egg-rr 74.4%

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t}} \cdot \ell \]

   if 2.26e-156 < k

   1. Initial program 49.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. *-lowering-*.f64 52.6

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{t}} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \]

      9. *-lowering-*.f64 60.7

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   7. Applied egg-rr 60.7%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.26 \cdot 10^{-156}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.7% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{+170}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 9.5e+170)
    (* l (/ l (* t_m (* (* t_m k) (* t_m k)))))
    (/ (* l l) (* t_m (* t_m (* t_m (* k k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9.5e+170) {
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	} else {
		tmp = (l * l) / (t_m * (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 9.5d+170) then
        tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))))
    else
        tmp = (l * l) / (t_m * (t_m * (t_m * (k * k))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9.5e+170) {
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	} else {
		tmp = (l * l) / (t_m * (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 9.5e+170:
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))))
	else:
		tmp = (l * l) / (t_m * (t_m * (t_m * (k * k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 9.5e+170)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(Float64(t_m * k) * Float64(t_m * k)))));
	else
		tmp = Float64(Float64(l * l) / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 9.5e+170)
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	else
		tmp = (l * l) / (t_m * (t_m * (t_m * (k * k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 9.5e+170], N[(l * N[(l / N[(t$95$m * N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] / N[(t$95$m * N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.5000000000000005e170

   1. Initial program 55.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. /-lowering-/.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. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. *-lowering-*.f64 53.3

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 53.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot k}} \cdot \ell \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right)} \cdot \ell \]

      11. *-lowering-*.f64 64.1

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot k\right)\right)} \cdot \ell \]

   7. Applied egg-rr 64.1%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}\right)} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]

      4. *-lowering-*.f64 66.2

         \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot t\right)\right)} \cdot \ell \]

   9. Applied egg-rr 66.2%

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(\left(k \cdot t\right) \cdot t\right)}} \cdot \ell \]

       2. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t}} \cdot \ell \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t}} \cdot \ell \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot t} \cdot \ell \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot t\right)\right) \cdot t} \cdot \ell \]

       6. *-lowering-*.f64 68.4

          \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot t} \cdot \ell \]

   11. Applied egg-rr 68.4%

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t}} \cdot \ell \]

   if 9.5000000000000005e170 < k

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. *-lowering-*.f64 54.8

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 54.8%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot t} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot t} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right) \cdot t} \]

      7. *-lowering-*.f64 65.4

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)\right) \cdot t} \]

   7. Applied egg-rr 65.4%

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right) \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{+170}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 65.0% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{t\_m \cdot \left(\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot k\right)\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* l (/ l (* t_m (* (* t_m k) (* t_m k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (t_m * ((t_m * k) * (t_m * k)))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (l * (l / (t_m * ((t_m * k) * (t_m * k)))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (t_m * ((t_m * k) * (t_m * k)))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * (l / (t_m * ((t_m * k) * (t_m * k)))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(l / Float64(t_m * Float64(Float64(t_m * k) * Float64(t_m * k))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (l * (l / (t_m * ((t_m * k) * (t_m * k)))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(l * N[(l / N[(t$95$m * N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. /-lowering-/.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. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   6. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   12. *-lowering-*.f64 53.5

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Simplified 53.5%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   5. associate-*r* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot k}} \cdot \ell \]

   6. *-commutative N/A

      \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

   8. associate-*l* N/A

      \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right)} \cdot \ell \]

   11. *-lowering-*.f64 63.2

       \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot k\right)\right)} \cdot \ell \]

7. Applied egg-rr 63.2%

   \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot \ell} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}\right)} \cdot \ell \]

   2. associate-*r* N/A

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]

   4. *-lowering-*.f64 65.1

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot t\right)\right)} \cdot \ell \]

9. Applied egg-rr 65.1%

   \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]
10. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(\left(k \cdot t\right) \cdot t\right)}} \cdot \ell \]

    2. associate-*r* N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t}} \cdot \ell \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t}} \cdot \ell \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot t} \cdot \ell \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot t\right)\right) \cdot t} \cdot \ell \]

    6. *-lowering-*.f64 67.0

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot t} \cdot \ell \]

11. Applied egg-rr 67.0%

    \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t}} \cdot \ell \]

12. Final simplification 67.0%

    \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)} \]
13. Add Preprocessing

Alternative 16: 62.7% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* l (/ l (* k (* t_m (* t_m (* t_m k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (k * (t_m * (t_m * (t_m * k))))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (l * (l / (k * (t_m * (t_m * (t_m * k))))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (k * (t_m * (t_m * (t_m * k))))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * (l / (k * (t_m * (t_m * (t_m * k))))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(l / Float64(k * Float64(t_m * Float64(t_m * Float64(t_m * k)))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (l * (l / (k * (t_m * (t_m * (t_m * k))))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(l * N[(l / N[(k * N[(t$95$m * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. /-lowering-/.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. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   6. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   12. *-lowering-*.f64 53.5

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Simplified 53.5%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   5. associate-*r* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot k}} \cdot \ell \]

   6. *-commutative N/A

      \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

   8. associate-*l* N/A

      \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right)} \cdot \ell \]

   11. *-lowering-*.f64 63.2

       \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot k\right)\right)} \cdot \ell \]

7. Applied egg-rr 63.2%

   \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot \ell} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}\right)} \cdot \ell \]

   2. associate-*r* N/A

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]

   4. *-lowering-*.f64 65.1

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot t\right)\right)} \cdot \ell \]

9. Applied egg-rr 65.1%

   \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]

10. Final simplification 65.1%

    \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(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))))