Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.2% → 94.2%

Time: 16.5s

Alternatives: 15

Speedup: 12.5×

• 27.0% 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: 54.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{+106}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(t\_2 \cdot \left(t\_m \cdot \left(t\_m \cdot \tan k\right)\right), t\_m \cdot 2, \left(\tan k \cdot \left(k \cdot t\_2\right)\right) \cdot \left(t\_m \cdot k\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \sin k\right)\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \end{array} \]

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 (/ (sin k) l)))
   (*
    t_s
    (if (<= t_m 5e+106)
      (/
       2.0
       (/
        (fma
         (* t_2 (* t_m (* t_m (tan k))))
         (* t_m 2.0)
         (* (* (tan k) (* k t_2)) (* t_m k)))
        l))
      (/
       2.0
       (*
        (* (tan k) (* (/ t_m l) (* (/ t_m l) (* t_m (sin k)))))
        (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = sin(k) / l;
	double tmp;
	if (t_m <= 5e+106) {
		tmp = 2.0 / (fma((t_2 * (t_m * (t_m * tan(k)))), (t_m * 2.0), ((tan(k) * (k * t_2)) * (t_m * k))) / l);
	} else {
		tmp = 2.0 / ((tan(k) * ((t_m / l) * ((t_m / l) * (t_m * sin(k))))) * (1.0 + (1.0 + pow((k / t_m), 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(sin(k) / l)
	tmp = 0.0
	if (t_m <= 5e+106)
		tmp = Float64(2.0 / Float64(fma(Float64(t_2 * Float64(t_m * Float64(t_m * tan(k)))), Float64(t_m * 2.0), Float64(Float64(tan(k) * Float64(k * t_2)) * Float64(t_m * k))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64(t_m / l) * Float64(Float64(t_m / l) * Float64(t_m * sin(k))))) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5e+106], N[(2.0 / N[(N[(N[(t$95$2 * N[(t$95$m * N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * 2.0), $MachinePrecision] + N[(N[(N[Tan[k], $MachinePrecision] * N[(k * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.9999999999999998e106

   1. Initial program 55.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 48.7%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 81.4%

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

   9. Applied rewrites 87.1%

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

   11. Applied rewrites 93.8%

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

   if 4.9999999999999998e106 < t

   1. Initial program 63.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 84.4

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

   4. Applied rewrites 84.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. lower-*.f64 90.3

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

   6. Applied rewrites 90.3%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{+106}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\frac{\sin k}{\ell} \cdot \left(t \cdot \left(t \cdot \tan k\right)\right), t \cdot 2, \left(\tan k \cdot \left(k \cdot \frac{\sin k}{\ell}\right)\right) \cdot \left(t \cdot k\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{+106}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(t\_m \cdot \left(\tan k \cdot \left(k \cdot t\_2\right)\right), k, t\_m \cdot \left(\left(t\_m \cdot \left(t\_m \cdot \tan k\right)\right) \cdot \left(2 \cdot t\_2\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \sin k\right)\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \end{array} \]

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 (/ (sin k) l)))
   (*
    t_s
    (if (<= t_m 5e+106)
      (/
       2.0
       (/
        (fma
         (* t_m (* (tan k) (* k t_2)))
         k
         (* t_m (* (* t_m (* t_m (tan k))) (* 2.0 t_2))))
        l))
      (/
       2.0
       (*
        (* (tan k) (* (/ t_m l) (* (/ t_m l) (* t_m (sin k)))))
        (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = sin(k) / l;
	double tmp;
	if (t_m <= 5e+106) {
		tmp = 2.0 / (fma((t_m * (tan(k) * (k * t_2))), k, (t_m * ((t_m * (t_m * tan(k))) * (2.0 * t_2)))) / l);
	} else {
		tmp = 2.0 / ((tan(k) * ((t_m / l) * ((t_m / l) * (t_m * sin(k))))) * (1.0 + (1.0 + pow((k / t_m), 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(sin(k) / l)
	tmp = 0.0
	if (t_m <= 5e+106)
		tmp = Float64(2.0 / Float64(fma(Float64(t_m * Float64(tan(k) * Float64(k * t_2))), k, Float64(t_m * Float64(Float64(t_m * Float64(t_m * tan(k))) * Float64(2.0 * t_2)))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64(t_m / l) * Float64(Float64(t_m / l) * Float64(t_m * sin(k))))) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5e+106], N[(2.0 / N[(N[(N[(t$95$m * N[(N[Tan[k], $MachinePrecision] * N[(k * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k + N[(t$95$m * N[(N[(t$95$m * N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.9999999999999998e106

   1. Initial program 55.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 48.7%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 81.4%

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

   9. Applied rewrites 87.1%

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

   11. Applied rewrites 93.8%

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

   if 4.9999999999999998e106 < t

   1. Initial program 63.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 84.4

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

   4. Applied rewrites 84.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. lower-*.f64 90.3

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

   6. Applied rewrites 90.3%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{+106}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(t \cdot \left(\tan k \cdot \left(k \cdot \frac{\sin k}{\ell}\right)\right), k, t \cdot \left(\left(t \cdot \left(t \cdot \tan k\right)\right) \cdot \left(2 \cdot \frac{\sin k}{\ell}\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{+106}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \mathsf{fma}\left(t\_2 \cdot \left(t\_m \cdot \left(t\_m \cdot \tan k\right)\right), 2, \left(k \cdot t\_2\right) \cdot \left(k \cdot \tan k\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \sin k\right)\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \end{array} \]

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 (/ (sin k) l)))
   (*
    t_s
    (if (<= t_m 5e+106)
      (/
       2.0
       (/
        (*
         t_m
         (fma (* t_2 (* t_m (* t_m (tan k)))) 2.0 (* (* k t_2) (* k (tan k)))))
        l))
      (/
       2.0
       (*
        (* (tan k) (* (/ t_m l) (* (/ t_m l) (* t_m (sin k)))))
        (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = sin(k) / l;
	double tmp;
	if (t_m <= 5e+106) {
		tmp = 2.0 / ((t_m * fma((t_2 * (t_m * (t_m * tan(k)))), 2.0, ((k * t_2) * (k * tan(k))))) / l);
	} else {
		tmp = 2.0 / ((tan(k) * ((t_m / l) * ((t_m / l) * (t_m * sin(k))))) * (1.0 + (1.0 + pow((k / t_m), 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(sin(k) / l)
	tmp = 0.0
	if (t_m <= 5e+106)
		tmp = Float64(2.0 / Float64(Float64(t_m * fma(Float64(t_2 * Float64(t_m * Float64(t_m * tan(k)))), 2.0, Float64(Float64(k * t_2) * Float64(k * tan(k))))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64(t_m / l) * Float64(Float64(t_m / l) * Float64(t_m * sin(k))))) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5e+106], N[(2.0 / N[(N[(t$95$m * N[(N[(t$95$2 * N[(t$95$m * N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(k * t$95$2), $MachinePrecision] * N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.9999999999999998e106

   1. Initial program 55.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 48.7%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 81.4%

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

   9. Applied rewrites 87.1%

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

   11. Applied rewrites 88.8%

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

   if 4.9999999999999998e106 < t

   1. Initial program 63.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 84.4

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

   4. Applied rewrites 84.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. lower-*.f64 90.3

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

   6. Applied rewrites 90.3%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{+106}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \mathsf{fma}\left(\frac{\sin k}{\ell} \cdot \left(t \cdot \left(t \cdot \tan k\right)\right), 2, \left(k \cdot \frac{\sin k}{\ell}\right) \cdot \left(k \cdot \tan k\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k}{\ell} \cdot \tan k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \mathsf{fma}\left(k, k \cdot t\_2, 2 \cdot \left(t\_2 \cdot \left(t\_m \cdot t\_m\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \sin k\right)\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \end{array} \]

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 (* (/ (sin k) l) (tan k))))
   (*
    t_s
    (if (<= t_m 7.2e+105)
      (/ 2.0 (/ (* t_m (fma k (* k t_2) (* 2.0 (* t_2 (* t_m t_m))))) l))
      (/
       2.0
       (*
        (* (tan k) (* (/ t_m l) (* (/ t_m l) (* t_m (sin k)))))
        (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (sin(k) / l) * tan(k);
	double tmp;
	if (t_m <= 7.2e+105) {
		tmp = 2.0 / ((t_m * fma(k, (k * t_2), (2.0 * (t_2 * (t_m * t_m))))) / l);
	} else {
		tmp = 2.0 / ((tan(k) * ((t_m / l) * ((t_m / l) * (t_m * sin(k))))) * (1.0 + (1.0 + pow((k / t_m), 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(sin(k) / l) * tan(k))
	tmp = 0.0
	if (t_m <= 7.2e+105)
		tmp = Float64(2.0 / Float64(Float64(t_m * fma(k, Float64(k * t_2), Float64(2.0 * Float64(t_2 * Float64(t_m * t_m))))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64(t_m / l) * Float64(Float64(t_m / l) * Float64(t_m * sin(k))))) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.2e+105], N[(2.0 / N[(N[(t$95$m * N[(k * N[(k * t$95$2), $MachinePrecision] + N[(2.0 * N[(t$95$2 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.1999999999999998e105

   1. Initial program 55.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 48.7%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 81.4%

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

   9. Applied rewrites 87.1%

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

   if 7.1999999999999998e105 < t

   1. Initial program 63.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 84.4

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

   4. Applied rewrites 84.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. lower-*.f64 90.3

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

   6. Applied rewrites 90.3%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \mathsf{fma}\left(k, k \cdot \left(\frac{\sin k}{\ell} \cdot \tan k\right), 2 \cdot \left(\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \left(t \cdot t\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.45 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \mathsf{fma}\left(k, k \cdot \left(\frac{\sin k}{\ell} \cdot \tan k\right), 2 \cdot \frac{k \cdot \left(k \cdot \left(t\_m \cdot t\_m\right)\right)}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \sin k\right)\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.45e-99)
    (/
     2.0
     (/
      (*
       t_m
       (fma
        k
        (* k (* (/ (sin k) l) (tan k)))
        (* 2.0 (/ (* k (* k (* t_m t_m))) l))))
      l))
    (/
     2.0
     (*
      (* (tan k) (* (/ t_m l) (* (/ t_m l) (* t_m (sin k)))))
      (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.45e-99) {
		tmp = 2.0 / ((t_m * fma(k, (k * ((sin(k) / l) * tan(k))), (2.0 * ((k * (k * (t_m * t_m))) / l)))) / l);
	} else {
		tmp = 2.0 / ((tan(k) * ((t_m / l) * ((t_m / l) * (t_m * sin(k))))) * (1.0 + (1.0 + pow((k / t_m), 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.45e-99)
		tmp = Float64(2.0 / Float64(Float64(t_m * fma(k, Float64(k * Float64(Float64(sin(k) / l) * tan(k))), Float64(2.0 * Float64(Float64(k * Float64(k * Float64(t_m * t_m))) / l)))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64(t_m / l) * Float64(Float64(t_m / l) * Float64(t_m * sin(k))))) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.45e-99], N[(2.0 / N[(N[(t$95$m * N[(k * N[(k * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(k * N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.44999999999999993e-99

   1. Initial program 53.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 42.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 81.6%

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

   9. Applied rewrites 86.1%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 79.9%

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

   if 1.44999999999999993e-99 < t

   1. Initial program 65.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 81.8

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

   4. Applied rewrites 81.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. lower-*.f64 85.1

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

   6. Applied rewrites 85.1%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.45 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \mathsf{fma}\left(k, k \cdot \left(\frac{\sin k}{\ell} \cdot \tan k\right), 2 \cdot \frac{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sin k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{-90}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \mathsf{fma}\left(k, k \cdot \left(\frac{\sin k}{\ell} \cdot \tan k\right), 2 \cdot \frac{k \cdot \left(k \cdot \left(t\_m \cdot t\_m\right)\right)}{\ell}\right)}{\ell}}\\ \mathbf{elif}\;t\_m \leq 1.95 \cdot 10^{+128}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot t\_m\right) \cdot \left(\frac{t\_2}{\ell} \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m \cdot t\_2}{\ell}\right)\right) \cdot 2}\\ \end{array} \end{array} \end{array} \]

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 4e-90)
      (/
       2.0
       (/
        (*
         t_m
         (fma
          k
          (* k (* (/ (sin k) l) (tan k)))
          (* 2.0 (/ (* k (* k (* t_m t_m))) l))))
        l))
      (if (<= t_m 1.95e+128)
        (/
         2.0
         (/
          (*
           (* t_m t_m)
           (* (/ t_2 l) (* (tan k) (fma k (/ k (* t_m t_m)) 2.0))))
          l))
        (/ 2.0 (* (* (tan k) (* (/ t_m l) (/ (* t_m t_2) l))) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * sin(k);
	double tmp;
	if (t_m <= 4e-90) {
		tmp = 2.0 / ((t_m * fma(k, (k * ((sin(k) / l) * tan(k))), (2.0 * ((k * (k * (t_m * t_m))) / l)))) / l);
	} else if (t_m <= 1.95e+128) {
		tmp = 2.0 / (((t_m * t_m) * ((t_2 / l) * (tan(k) * fma(k, (k / (t_m * t_m)), 2.0)))) / l);
	} else {
		tmp = 2.0 / ((tan(k) * ((t_m / l) * ((t_m * t_2) / l))) * 2.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m * sin(k))
	tmp = 0.0
	if (t_m <= 4e-90)
		tmp = Float64(2.0 / Float64(Float64(t_m * fma(k, Float64(k * Float64(Float64(sin(k) / l) * tan(k))), Float64(2.0 * Float64(Float64(k * Float64(k * Float64(t_m * t_m))) / l)))) / l));
	elseif (t_m <= 1.95e+128)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * t_m) * Float64(Float64(t_2 / l) * Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64(t_m / l) * Float64(Float64(t_m * t_2) / l))) * 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4e-90], N[(2.0 / N[(N[(t$95$m * N[(k * N[(k * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(k * N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.95e+128], N[(2.0 / N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(N[(t$95$2 / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * t$95$2), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3.99999999999999998e-90

   1. Initial program 53.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 42.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 81.6%

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

   9. Applied rewrites 86.1%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 79.9%

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

   if 3.99999999999999998e-90 < t < 1.9499999999999999e128

   1. Initial program 64.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 78.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lift-pow.f64 N/A

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

   6. Applied rewrites 87.3%

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

   if 1.9499999999999999e128 < t

   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. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 83.0

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

   4. Applied rewrites 83.0%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 83.0%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-90}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \mathsf{fma}\left(k, k \cdot \left(\frac{\sin k}{\ell} \cdot \tan k\right), 2 \cdot \frac{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}{\ell}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{+128}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot t\right) \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right)\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.3 \cdot 10^{+108}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \left(t\_m \cdot \left(\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m \cdot \left(t\_m \cdot \sin k\right)}{\ell}\right)\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.3e+108)
    (/
     2.0
     (*
      (/ 1.0 l)
      (* t_m (* (* (/ (sin k) l) (tan k)) (fma 2.0 (* t_m t_m) (* k k))))))
    (/ 2.0 (* (* (tan k) (* (/ t_m l) (/ (* t_m (* t_m (sin k))) l))) 2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.3e+108) {
		tmp = 2.0 / ((1.0 / l) * (t_m * (((sin(k) / l) * tan(k)) * fma(2.0, (t_m * t_m), (k * k)))));
	} else {
		tmp = 2.0 / ((tan(k) * ((t_m / l) * ((t_m * (t_m * sin(k))) / l))) * 2.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.3e+108)
		tmp = Float64(2.0 / Float64(Float64(1.0 / l) * Float64(t_m * Float64(Float64(Float64(sin(k) / l) * tan(k)) * fma(2.0, Float64(t_m * t_m), Float64(k * k))))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64(t_m / l) * Float64(Float64(t_m * Float64(t_m * sin(k))) / l))) * 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.3e+108], N[(2.0 / N[(N[(1.0 / l), $MachinePrecision] * N[(t$95$m * N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.30000000000000019e108

   1. Initial program 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 81.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 81.6

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

   9. Applied rewrites 83.6%

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

   if 3.30000000000000019e108 < t

   1. Initial program 66.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 83.8

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

   4. Applied rewrites 83.8%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 83.8%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{+108}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \left(t \cdot \left(\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right)\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.3 \cdot 10^{+108}:\\ \;\;\;\;\frac{2 \cdot \ell}{t\_m \cdot \left(\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m \cdot \left(t\_m \cdot \sin k\right)}{\ell}\right)\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.3e+108)
    (/
     (* 2.0 l)
     (* t_m (* (* (/ (sin k) l) (tan k)) (fma 2.0 (* t_m t_m) (* k k)))))
    (/ 2.0 (* (* (tan k) (* (/ t_m l) (/ (* t_m (* t_m (sin k))) l))) 2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.3e+108) {
		tmp = (2.0 * l) / (t_m * (((sin(k) / l) * tan(k)) * fma(2.0, (t_m * t_m), (k * k))));
	} else {
		tmp = 2.0 / ((tan(k) * ((t_m / l) * ((t_m * (t_m * sin(k))) / l))) * 2.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.3e+108)
		tmp = Float64(Float64(2.0 * l) / Float64(t_m * Float64(Float64(Float64(sin(k) / l) * tan(k)) * fma(2.0, Float64(t_m * t_m), Float64(k * k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64(t_m / l) * Float64(Float64(t_m * Float64(t_m * sin(k))) / l))) * 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.3e+108], N[(N[(2.0 * l), $MachinePrecision] / N[(t$95$m * N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.30000000000000019e108

   1. Initial program 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 81.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l/ N/A

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

   9. Applied rewrites 83.6%

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

   if 3.30000000000000019e108 < t

   1. Initial program 66.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 83.8

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

   4. Applied rewrites 83.8%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 83.8%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{+108}:\\ \;\;\;\;\frac{2 \cdot \ell}{t \cdot \left(\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right)\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.3 \cdot 10^{+108}:\\ \;\;\;\;\ell \cdot \frac{2}{t\_m \cdot \left(\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m \cdot \left(t\_m \cdot \sin k\right)}{\ell}\right)\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.3e+108)
    (*
     l
     (/
      2.0
      (* t_m (* (* (/ (sin k) l) (tan k)) (fma 2.0 (* t_m t_m) (* k k))))))
    (/ 2.0 (* (* (tan k) (* (/ t_m l) (/ (* t_m (* t_m (sin k))) l))) 2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.3e+108) {
		tmp = l * (2.0 / (t_m * (((sin(k) / l) * tan(k)) * fma(2.0, (t_m * t_m), (k * k)))));
	} else {
		tmp = 2.0 / ((tan(k) * ((t_m / l) * ((t_m * (t_m * sin(k))) / l))) * 2.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.3e+108)
		tmp = Float64(l * Float64(2.0 / Float64(t_m * Float64(Float64(Float64(sin(k) / l) * tan(k)) * fma(2.0, Float64(t_m * t_m), Float64(k * k))))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64(t_m / l) * Float64(Float64(t_m * Float64(t_m * sin(k))) / l))) * 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.3e+108], N[(l * N[(2.0 / N[(t$95$m * N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.30000000000000019e108

   1. Initial program 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 81.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   9. Applied rewrites 83.4%

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

   if 3.30000000000000019e108 < t

   1. Initial program 66.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 83.8

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

   4. Applied rewrites 83.8%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 83.8%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{+108}:\\ \;\;\;\;\ell \cdot \frac{2}{t \cdot \left(\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right)\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{+108}:\\ \;\;\;\;\ell \cdot \frac{2}{t\_m \cdot \left(\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 1.8e+108)
    (*
     l
     (/
      2.0
      (* t_m (* (* (/ (sin k) l) (tan k)) (fma 2.0 (* t_m t_m) (* k k))))))
    (* 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 <= 1.8e+108) {
		tmp = l * (2.0 / (t_m * (((sin(k) / l) * tan(k)) * fma(2.0, (t_m * t_m), (k * k)))));
	} 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 <= 1.8e+108)
		tmp = Float64(l * Float64(2.0 / Float64(t_m * Float64(Float64(Float64(sin(k) / l) * tan(k)) * fma(2.0, Float64(t_m * t_m), Float64(k * k))))));
	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

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, 1.8e+108], N[(l * N[(2.0 / N[(t$95$m * N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $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 2 regimes

2. if t < 1.8e108

   1. Initial program 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 81.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   9. Applied rewrites 83.4%

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

   if 1.8e108 < t

   1. Initial program 66.0%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 52.8

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

   5. Applied rewrites 52.8%

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

   7. Applied rewrites 75.3%

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

   9. Applied rewrites 75.3%

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

   11. Applied rewrites 83.8%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{+108}:\\ \;\;\;\;\ell \cdot \frac{2}{t \cdot \left(\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.4% accurate, 4.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot t\_m}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.5 \cdot 10^{+56}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(k \cdot \left(k \cdot \mathsf{fma}\left(k, k \cdot \mathsf{fma}\left(t\_2, 0.3333333333333333, \frac{1}{\ell}\right), 2 \cdot t\_2\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot k\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (* t_m t_m) l)))
   (*
    t_s
    (if (<= t_m 3.5e+56)
      (/
       2.0
       (/
        (*
         t_m
         (*
          k
          (*
           k
           (fma k (* k (fma t_2 0.3333333333333333 (/ 1.0 l))) (* 2.0 t_2)))))
        l))
      (* 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 t_2 = (t_m * t_m) / l;
	double tmp;
	if (t_m <= 3.5e+56) {
		tmp = 2.0 / ((t_m * (k * (k * fma(k, (k * fma(t_2, 0.3333333333333333, (1.0 / l))), (2.0 * t_2))))) / l);
	} 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)
	t_2 = Float64(Float64(t_m * t_m) / l)
	tmp = 0.0
	if (t_m <= 3.5e+56)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(k * Float64(k * fma(k, Float64(k * fma(t_2, 0.3333333333333333, Float64(1.0 / l))), Float64(2.0 * t_2))))) / l));
	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

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 * t$95$m), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.5e+56], N[(2.0 / N[(N[(t$95$m * N[(k * N[(k * N[(k * N[(k * N[(t$95$2 * 0.3333333333333333 + N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $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 2 regimes

2. if t < 3.49999999999999999e56

   1. Initial program 54.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 81.4%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 72.0%

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

   if 3.49999999999999999e56 < t

   1. Initial program 65.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 56.0

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

   5. Applied rewrites 56.0%

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

   7. Applied rewrites 75.1%

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

   9. Applied rewrites 75.1%

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

   11. Applied rewrites 81.1%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{+56}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(k \cdot \left(k \cdot \mathsf{fma}\left(k, k \cdot \mathsf{fma}\left(\frac{t \cdot t}{\ell}, 0.3333333333333333, \frac{1}{\ell}\right), 2 \cdot \frac{t \cdot t}{\ell}\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.9% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-162}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 2.5e-162)
    (* 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 <= 2.5e-162) {
		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 <= 2.5d-162) 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 <= 2.5e-162) {
		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 <= 2.5e-162:
		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 <= 2.5e-162)
		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 <= 2.5e-162)
		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, 2.5e-162], 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 < 2.50000000000000007e-162

   1. Initial program 60.8%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 54.7

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

   5. Applied rewrites 54.7%

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

   7. Applied rewrites 70.0%

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

   9. Applied rewrites 70.0%

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

   11. Applied rewrites 74.0%

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

   if 2.50000000000000007e-162 < k

   1. Initial program 52.5%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 53.0

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

   5. Applied rewrites 53.0%

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

   7. Applied rewrites 65.8%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-162}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.5% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{+195}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

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 3.8e+195)
    (* 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 <= 3.8e+195) {
		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 <= 3.8d+195) 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 <= 3.8e+195) {
		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 <= 3.8e+195:
		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 <= 3.8e+195)
		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 <= 3.8e+195)
		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, 3.8e+195], 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 < 3.8e195

   1. Initial program 57.4%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 53.7

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

   5. Applied rewrites 53.7%

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

   7. Applied rewrites 64.8%

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

   9. Applied rewrites 64.8%

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

   11. Applied rewrites 67.8%

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

   if 3.8e195 < k

   1. Initial program 55.3%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 55.5

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

   5. Applied rewrites 55.5%

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

   7. Applied rewrites 79.4%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{+195}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 66.0% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{t\_m \cdot \left(\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot k\right)\right)}\right) \end{array} \]

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 57.2%

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

3. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. cube-mult N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 53.9

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

5. Applied rewrites 53.9%

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

7. Applied rewrites 63.9%

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

9. Applied rewrites 64.3%

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

11. Applied rewrites 67.4%

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

12. Final simplification 67.4%

    \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)} \]
13. Add Preprocessing

Alternative 15: 63.6% 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 57.2%

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

3. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. cube-mult N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 53.9

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

5. Applied rewrites 53.9%

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

7. Applied rewrites 63.9%

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

9. Applied rewrites 64.3%

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

10. Final simplification 64.3%

    \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024226 
(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))))