Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.5% → 93.8%

Time: 19.2s

Alternatives: 22

Speedup: 12.5×

• 27.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.5% 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: 93.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{+20}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(t\_m \cdot \left(t\_2 \cdot \left(k \cdot \tan k\right)\right), k, \left(t\_2 \cdot \tan k\right) \cdot \left(2 \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right)\right)\right)}{\ell}}\\ \mathbf{elif}\;t\_m \leq 9.2 \cdot 10^{+187}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \left(t\_m \cdot t\_2\right)\right) \cdot \frac{t\_m \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t\_m \cdot \left(\frac{t\_m \cdot \sin k}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)\right) \cdot \left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1\right)}\\ \end{array} \end{array} \end{array} \]

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 2.9e+20)
      (/
       2.0
       (/
        (fma
         (* t_m (* t_2 (* k (tan k))))
         k
         (* (* t_2 (tan k)) (* 2.0 (* t_m (* t_m t_m)))))
        l))
      (if (<= t_m 9.2e+187)
        (/
         2.0
         (*
          (* t_m (* t_m t_2))
          (/ (* t_m (* (tan k) (fma k (/ k (* t_m t_m)) 2.0))) l)))
        (/
         2.0
         (*
          (* (tan k) (* t_m (* (/ (* t_m (sin k)) l) (/ t_m l))))
          (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0))))))))

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 <= 2.9e+20) {
		tmp = 2.0 / (fma((t_m * (t_2 * (k * tan(k)))), k, ((t_2 * tan(k)) * (2.0 * (t_m * (t_m * t_m))))) / l);
	} else if (t_m <= 9.2e+187) {
		tmp = 2.0 / ((t_m * (t_m * t_2)) * ((t_m * (tan(k) * fma(k, (k / (t_m * t_m)), 2.0))) / l));
	} else {
		tmp = 2.0 / ((tan(k) * (t_m * (((t_m * sin(k)) / l) * (t_m / l)))) * ((pow((k / t_m), 2.0) + 1.0) + 1.0));
	}
	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 <= 2.9e+20)
		tmp = Float64(2.0 / Float64(fma(Float64(t_m * Float64(t_2 * Float64(k * tan(k)))), k, Float64(Float64(t_2 * tan(k)) * Float64(2.0 * Float64(t_m * Float64(t_m * t_m))))) / l));
	elseif (t_m <= 9.2e+187)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(t_m * t_2)) * Float64(Float64(t_m * 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(t_m * Float64(Float64(Float64(t_m * sin(k)) / l) * Float64(t_m / l)))) * Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0)));
	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, 2.9e+20], N[(2.0 / N[(N[(N[(t$95$m * N[(t$95$2 * N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k + N[(N[(t$95$2 * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.2e+187], N[(2.0 / N[(N[(t$95$m * N[(t$95$m * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.9e20

   1. Initial program 55.1%

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

   4. Applied rewrites 47.2%

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

   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. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 80.5%

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

   9. Applied rewrites 87.2%

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

       1. lift-tan.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-tan.f64 N/A

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

       9. lift-sin.f64 N/A

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

       10. lift-/.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. lift-*.f64 N/A

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

   11. Applied rewrites 89.2%

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

   if 2.9e20 < t < 9.20000000000000015e187

   1. Initial program 51.6%

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

   4. Applied rewrites 59.0%

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

   6. Applied rewrites 71.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 99.3%

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

   if 9.20000000000000015e187 < t

   1. Initial program 52.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 56.7

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

   4. Applied rewrites 56.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*l/ N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 88.5

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

   6. Applied rewrites 88.5%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{+20}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(t \cdot \left(\frac{\sin k}{\ell} \cdot \left(k \cdot \tan k\right)\right), k, \left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \left(2 \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)\right)}{\ell}}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{+187}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(t \cdot \frac{\sin k}{\ell}\right)\right) \cdot \frac{t \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k}{\ell}\\ t_3 := t\_2 \cdot \tan k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{+20}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \mathsf{fma}\left(k, k \cdot t\_3, t\_3 \cdot \left(2 \cdot \left(t\_m \cdot t\_m\right)\right)\right)}{\ell}}\\ \mathbf{elif}\;t\_m \leq 9.2 \cdot 10^{+187}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \left(t\_m \cdot t\_2\right)\right) \cdot \frac{t\_m \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t\_m \cdot \left(\frac{t\_m \cdot \sin k}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)\right) \cdot \left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1\right)}\\ \end{array} \end{array} \end{array} \]

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_3 (* t_2 (tan k))))
   (*
    t_s
    (if (<= t_m 2.9e+20)
      (/ 2.0 (/ (* t_m (fma k (* k t_3) (* t_3 (* 2.0 (* t_m t_m))))) l))
      (if (<= t_m 9.2e+187)
        (/
         2.0
         (*
          (* t_m (* t_m t_2))
          (/ (* t_m (* (tan k) (fma k (/ k (* t_m t_m)) 2.0))) l)))
        (/
         2.0
         (*
          (* (tan k) (* t_m (* (/ (* t_m (sin k)) l) (/ t_m l))))
          (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0))))))))

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 t_3 = t_2 * tan(k);
	double tmp;
	if (t_m <= 2.9e+20) {
		tmp = 2.0 / ((t_m * fma(k, (k * t_3), (t_3 * (2.0 * (t_m * t_m))))) / l);
	} else if (t_m <= 9.2e+187) {
		tmp = 2.0 / ((t_m * (t_m * t_2)) * ((t_m * (tan(k) * fma(k, (k / (t_m * t_m)), 2.0))) / l));
	} else {
		tmp = 2.0 / ((tan(k) * (t_m * (((t_m * sin(k)) / l) * (t_m / l)))) * ((pow((k / t_m), 2.0) + 1.0) + 1.0));
	}
	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)
	t_3 = Float64(t_2 * tan(k))
	tmp = 0.0
	if (t_m <= 2.9e+20)
		tmp = Float64(2.0 / Float64(Float64(t_m * fma(k, Float64(k * t_3), Float64(t_3 * Float64(2.0 * Float64(t_m * t_m))))) / l));
	elseif (t_m <= 9.2e+187)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(t_m * t_2)) * Float64(Float64(t_m * 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(t_m * Float64(Float64(Float64(t_m * sin(k)) / l) * Float64(t_m / l)))) * Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0)));
	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]}, Block[{t$95$3 = N[(t$95$2 * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.9e+20], N[(2.0 / N[(N[(t$95$m * N[(k * N[(k * t$95$3), $MachinePrecision] + N[(t$95$3 * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.2e+187], N[(2.0 / N[(N[(t$95$m * N[(t$95$m * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.9e20

   1. Initial program 55.1%

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

   4. Applied rewrites 47.2%

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

   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. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 80.5%

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

   9. Applied rewrites 87.2%

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

   if 2.9e20 < t < 9.20000000000000015e187

   1. Initial program 51.6%

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

   4. Applied rewrites 59.0%

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

   6. Applied rewrites 71.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 99.3%

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

   if 9.20000000000000015e187 < t

   1. Initial program 52.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 56.7

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

   4. Applied rewrites 56.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*l/ N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 88.5

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

   6. Applied rewrites 88.5%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{+20}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \mathsf{fma}\left(k, k \cdot \left(\frac{\sin k}{\ell} \cdot \tan k\right), \left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \left(2 \cdot \left(t \cdot t\right)\right)\right)}{\ell}}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{+187}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(t \cdot \frac{\sin k}{\ell}\right)\right) \cdot \frac{t \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k}{\ell}\\ t_3 := t\_2 \cdot \tan k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{-46}:\\ \;\;\;\;\frac{2 \cdot \ell}{t\_m \cdot \mathsf{fma}\left(k \cdot k, t\_3, t\_3 \cdot \left(2 \cdot \left(t\_m \cdot t\_m\right)\right)\right)}\\ \mathbf{elif}\;t\_m \leq 9.2 \cdot 10^{+187}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \left(t\_m \cdot t\_2\right)\right) \cdot \frac{t\_m \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t\_m \cdot \left(\frac{t\_m \cdot \sin k}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)\right) \cdot \left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1\right)}\\ \end{array} \end{array} \end{array} \]

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_3 (* t_2 (tan k))))
   (*
    t_s
    (if (<= t_m 2.9e-46)
      (/ (* 2.0 l) (* t_m (fma (* k k) t_3 (* t_3 (* 2.0 (* t_m t_m))))))
      (if (<= t_m 9.2e+187)
        (/
         2.0
         (*
          (* t_m (* t_m t_2))
          (/ (* t_m (* (tan k) (fma k (/ k (* t_m t_m)) 2.0))) l)))
        (/
         2.0
         (*
          (* (tan k) (* t_m (* (/ (* t_m (sin k)) l) (/ t_m l))))
          (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0))))))))

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 t_3 = t_2 * tan(k);
	double tmp;
	if (t_m <= 2.9e-46) {
		tmp = (2.0 * l) / (t_m * fma((k * k), t_3, (t_3 * (2.0 * (t_m * t_m)))));
	} else if (t_m <= 9.2e+187) {
		tmp = 2.0 / ((t_m * (t_m * t_2)) * ((t_m * (tan(k) * fma(k, (k / (t_m * t_m)), 2.0))) / l));
	} else {
		tmp = 2.0 / ((tan(k) * (t_m * (((t_m * sin(k)) / l) * (t_m / l)))) * ((pow((k / t_m), 2.0) + 1.0) + 1.0));
	}
	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)
	t_3 = Float64(t_2 * tan(k))
	tmp = 0.0
	if (t_m <= 2.9e-46)
		tmp = Float64(Float64(2.0 * l) / Float64(t_m * fma(Float64(k * k), t_3, Float64(t_3 * Float64(2.0 * Float64(t_m * t_m))))));
	elseif (t_m <= 9.2e+187)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(t_m * t_2)) * Float64(Float64(t_m * 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(t_m * Float64(Float64(Float64(t_m * sin(k)) / l) * Float64(t_m / l)))) * Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0)));
	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]}, Block[{t$95$3 = N[(t$95$2 * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.9e-46], N[(N[(2.0 * l), $MachinePrecision] / N[(t$95$m * N[(N[(k * k), $MachinePrecision] * t$95$3 + N[(t$95$3 * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.2e+187], N[(2.0 / N[(N[(t$95$m * N[(t$95$m * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.90000000000000005e-46

   1. Initial program 54.4%

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

   4. Applied rewrites 45.6%

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

   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. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 81.4%

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

   9. Applied rewrites 85.3%

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

   if 2.90000000000000005e-46 < t < 9.20000000000000015e187

   1. Initial program 54.8%

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

   4. Applied rewrites 60.5%

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

   6. Applied rewrites 70.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 93.8%

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

   if 9.20000000000000015e187 < t

   1. Initial program 52.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 56.7

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

   4. Applied rewrites 56.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*l/ N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 88.5

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

   6. Applied rewrites 88.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{-46}:\\ \;\;\;\;\frac{2 \cdot \ell}{t \cdot \mathsf{fma}\left(k \cdot k, \frac{\sin k}{\ell} \cdot \tan k, \left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \left(2 \cdot \left(t \cdot t\right)\right)\right)}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{+187}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(t \cdot \frac{\sin k}{\ell}\right)\right) \cdot \frac{t \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(\left(t\_2 \cdot \tan k\right) \cdot \mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right)\right)}{\ell}}\\ \mathbf{elif}\;t\_m \leq 9.2 \cdot 10^{+187}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \left(t\_m \cdot t\_2\right)\right) \cdot \frac{t\_m \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t\_m \cdot \left(\frac{t\_m \cdot \sin k}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)\right) \cdot \left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1\right)}\\ \end{array} \end{array} \end{array} \]

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 2.9e-46)
      (/ 2.0 (/ (* t_m (* (* t_2 (tan k)) (fma 2.0 (* t_m t_m) (* k k)))) l))
      (if (<= t_m 9.2e+187)
        (/
         2.0
         (*
          (* t_m (* t_m t_2))
          (/ (* t_m (* (tan k) (fma k (/ k (* t_m t_m)) 2.0))) l)))
        (/
         2.0
         (*
          (* (tan k) (* t_m (* (/ (* t_m (sin k)) l) (/ t_m l))))
          (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0))))))))

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 <= 2.9e-46) {
		tmp = 2.0 / ((t_m * ((t_2 * tan(k)) * fma(2.0, (t_m * t_m), (k * k)))) / l);
	} else if (t_m <= 9.2e+187) {
		tmp = 2.0 / ((t_m * (t_m * t_2)) * ((t_m * (tan(k) * fma(k, (k / (t_m * t_m)), 2.0))) / l));
	} else {
		tmp = 2.0 / ((tan(k) * (t_m * (((t_m * sin(k)) / l) * (t_m / l)))) * ((pow((k / t_m), 2.0) + 1.0) + 1.0));
	}
	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 <= 2.9e-46)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(t_2 * tan(k)) * fma(2.0, Float64(t_m * t_m), Float64(k * k)))) / l));
	elseif (t_m <= 9.2e+187)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(t_m * t_2)) * Float64(Float64(t_m * 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(t_m * Float64(Float64(Float64(t_m * sin(k)) / l) * Float64(t_m / l)))) * Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0)));
	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, 2.9e-46], N[(2.0 / N[(N[(t$95$m * N[(N[(t$95$2 * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.2e+187], N[(2.0 / N[(N[(t$95$m * N[(t$95$m * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.90000000000000005e-46

   1. Initial program 54.4%

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

   4. Applied rewrites 45.6%

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

   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. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 81.4%

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

   9. Applied rewrites 87.3%

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

       1. lift-tan.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-tan.f64 N/A

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

       9. lift-sin.f64 N/A

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

       10. lift-/.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-fma.f64 N/A

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

   11. Applied rewrites 85.1%

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

   if 2.90000000000000005e-46 < t < 9.20000000000000015e187

   1. Initial program 54.8%

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

   4. Applied rewrites 60.5%

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

   6. Applied rewrites 70.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 93.8%

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

   if 9.20000000000000015e187 < t

   1. Initial program 52.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 56.7

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

   4. Applied rewrites 56.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*l/ N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 88.5

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

   6. Applied rewrites 88.5%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{+187}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(t \cdot \frac{\sin k}{\ell}\right)\right) \cdot \frac{t \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k}{\ell}\\ t_3 := t\_m \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(\left(t\_2 \cdot \tan k\right) \cdot \mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right)\right)}{\ell}}\\ \mathbf{elif}\;t\_m \leq 10^{+147}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \left(t\_m \cdot t\_2\right)\right) \cdot \frac{t\_3}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_3 \cdot \frac{\left(t\_m \cdot \sin k\right) \cdot \frac{t\_m}{\ell}}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (sin k) l))
        (t_3 (* t_m (* (tan k) (fma k (/ k (* t_m t_m)) 2.0)))))
   (*
    t_s
    (if (<= t_m 2.9e-46)
      (/ 2.0 (/ (* t_m (* (* t_2 (tan k)) (fma 2.0 (* t_m t_m) (* k k)))) l))
      (if (<= t_m 1e+147)
        (/ 2.0 (* (* t_m (* t_m t_2)) (/ t_3 l)))
        (/ 2.0 (* t_3 (/ (* (* t_m (sin k)) (/ t_m l)) l))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = sin(k) / l;
	double t_3 = t_m * (tan(k) * fma(k, (k / (t_m * t_m)), 2.0));
	double tmp;
	if (t_m <= 2.9e-46) {
		tmp = 2.0 / ((t_m * ((t_2 * tan(k)) * fma(2.0, (t_m * t_m), (k * k)))) / l);
	} else if (t_m <= 1e+147) {
		tmp = 2.0 / ((t_m * (t_m * t_2)) * (t_3 / l));
	} else {
		tmp = 2.0 / (t_3 * (((t_m * sin(k)) * (t_m / l)) / l));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(sin(k) / l)
	t_3 = Float64(t_m * Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)))
	tmp = 0.0
	if (t_m <= 2.9e-46)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(t_2 * tan(k)) * fma(2.0, Float64(t_m * t_m), Float64(k * k)))) / l));
	elseif (t_m <= 1e+147)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(t_m * t_2)) * Float64(t_3 / l)));
	else
		tmp = Float64(2.0 / Float64(t_3 * Float64(Float64(Float64(t_m * sin(k)) * Float64(t_m / l)) / l)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.9e-46], N[(2.0 / N[(N[(t$95$m * N[(N[(t$95$2 * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1e+147], N[(2.0 / N[(N[(t$95$m * N[(t$95$m * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$3 * N[(N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.90000000000000005e-46

   1. Initial program 54.4%

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

   4. Applied rewrites 45.6%

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

   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. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 81.4%

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

   9. Applied rewrites 87.3%

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

       1. lift-tan.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-tan.f64 N/A

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

       9. lift-sin.f64 N/A

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

       10. lift-/.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-fma.f64 N/A

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

   11. Applied rewrites 85.1%

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

   if 2.90000000000000005e-46 < t < 9.9999999999999998e146

   1. Initial program 57.0%

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

   4. Applied rewrites 61.7%

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

   6. Applied rewrites 66.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 92.3%

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

   if 9.9999999999999998e146 < t

   1. Initial program 50.9%

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

   4. Applied rewrites 48.2%

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

   6. Applied rewrites 69.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. times-frac N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 89.5

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

   8. Applied rewrites 89.5%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}}\\ \mathbf{elif}\;t \leq 10^{+147}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(t \cdot \frac{\sin k}{\ell}\right)\right) \cdot \frac{t \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)\right) \cdot \frac{\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(\left(t\_2 \cdot \tan k\right) \cdot \mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right)\right)}{\ell}}\\ \mathbf{elif}\;t\_m \leq 1.2 \cdot 10^{+219}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \left(t\_m \cdot t\_2\right)\right) \cdot \frac{t\_m \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\frac{\ell}{t\_m}}{t\_m \cdot k}\\ \end{array} \end{array} \end{array} \]

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 2.9e-46)
      (/ 2.0 (/ (* t_m (* (* t_2 (tan k)) (fma 2.0 (* t_m t_m) (* k k)))) l))
      (if (<= t_m 1.2e+219)
        (/
         2.0
         (*
          (* t_m (* t_m t_2))
          (/ (* t_m (* (tan k) (fma k (/ k (* t_m t_m)) 2.0))) l)))
        (* (/ l (* t_m k)) (/ (/ l t_m) (* t_m k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = sin(k) / l;
	double tmp;
	if (t_m <= 2.9e-46) {
		tmp = 2.0 / ((t_m * ((t_2 * tan(k)) * fma(2.0, (t_m * t_m), (k * k)))) / l);
	} else if (t_m <= 1.2e+219) {
		tmp = 2.0 / ((t_m * (t_m * t_2)) * ((t_m * (tan(k) * fma(k, (k / (t_m * t_m)), 2.0))) / l));
	} else {
		tmp = (l / (t_m * k)) * ((l / t_m) / (t_m * k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(sin(k) / l)
	tmp = 0.0
	if (t_m <= 2.9e-46)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(t_2 * tan(k)) * fma(2.0, Float64(t_m * t_m), Float64(k * k)))) / l));
	elseif (t_m <= 1.2e+219)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(t_m * t_2)) * Float64(Float64(t_m * Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0))) / l)));
	else
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(Float64(l / t_m) / Float64(t_m * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.9e-46], N[(2.0 / N[(N[(t$95$m * N[(N[(t$95$2 * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.2e+219], N[(2.0 / N[(N[(t$95$m * N[(t$95$m * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.90000000000000005e-46

   1. Initial program 54.4%

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

   4. Applied rewrites 45.6%

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

   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. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 81.4%

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

   9. Applied rewrites 87.3%

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

       1. lift-tan.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-tan.f64 N/A

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

       9. lift-sin.f64 N/A

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

       10. lift-/.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-fma.f64 N/A

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

   11. Applied rewrites 85.1%

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

   if 2.90000000000000005e-46 < t < 1.2e219

   1. Initial program 51.8%

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

   4. Applied rewrites 57.2%

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

   6. Applied rewrites 66.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 90.7%

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

   if 1.2e219 < t

   1. Initial program 59.9%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-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 45.5

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

   5. Applied rewrites 45.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. unswap-sqr N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 91.3

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

   7. Applied rewrites 91.3%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+219}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(t \cdot \frac{\sin k}{\ell}\right)\right) \cdot \frac{t \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\frac{\ell}{t}}{t \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.85 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right)\right)}{\ell}}\\ \mathbf{elif}\;t\_m \leq 3 \cdot 10^{+165}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left(t\_m \cdot \frac{t\_m \cdot \sin k}{\ell}\right)} \cdot \frac{\ell}{\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{k \cdot \left(t\_m \cdot \frac{t\_m \cdot k}{\ell}\right)}\\ \end{array} \end{array} \]

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 2.85e-46)
    (/
     2.0
     (/ (* t_m (* (* (/ (sin k) l) (tan k)) (fma 2.0 (* t_m t_m) (* k k)))) l))
    (if (<= t_m 3e+165)
      (*
       (/ 2.0 (* t_m (* t_m (/ (* t_m (sin k)) l))))
       (/ l (* (tan k) (fma k (/ k (* t_m t_m)) 2.0))))
      (/ (/ l t_m) (* k (* t_m (/ (* t_m k) l))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.85e-46) {
		tmp = 2.0 / ((t_m * (((sin(k) / l) * tan(k)) * fma(2.0, (t_m * t_m), (k * k)))) / l);
	} else if (t_m <= 3e+165) {
		tmp = (2.0 / (t_m * (t_m * ((t_m * sin(k)) / l)))) * (l / (tan(k) * fma(k, (k / (t_m * t_m)), 2.0)));
	} else {
		tmp = (l / t_m) / (k * (t_m * ((t_m * k) / l)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.85e-46)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(Float64(sin(k) / l) * tan(k)) * fma(2.0, Float64(t_m * t_m), Float64(k * k)))) / l));
	elseif (t_m <= 3e+165)
		tmp = Float64(Float64(2.0 / Float64(t_m * Float64(t_m * Float64(Float64(t_m * sin(k)) / l)))) * Float64(l / Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0))));
	else
		tmp = Float64(Float64(l / t_m) / Float64(k * Float64(t_m * Float64(Float64(t_m * k) / l))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.85e-46], N[(2.0 / N[(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] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3e+165], N[(N[(2.0 / N[(t$95$m * N[(t$95$m * N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] / N[(k * N[(t$95$m * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.8500000000000001e-46

   1. Initial program 54.4%

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

   4. Applied rewrites 45.6%

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

   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. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 81.4%

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

   9. Applied rewrites 87.3%

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

       1. lift-tan.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-tan.f64 N/A

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

       9. lift-sin.f64 N/A

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

       10. lift-/.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-fma.f64 N/A

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

   11. Applied rewrites 85.1%

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

   if 2.8500000000000001e-46 < t < 2.9999999999999999e165

   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. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 30.6

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

   4. Applied rewrites 30.6%

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

   6. Applied rewrites 91.1%

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

   if 2.9999999999999999e165 < t

   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 41.9

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

   5. Applied rewrites 41.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 46.5

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 46.5

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

   7. Applied rewrites 46.5%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 72.2

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

   9. Applied rewrites 72.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/r* N/A

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

       5. lift-/.f64 N/A

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

       6. associate-*l/ N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. frac-times N/A

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

       10. lift-/.f64 N/A

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

       11. clear-num N/A

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

       12. un-div-inv N/A

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

       13. lower-/.f64 N/A

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

       14. div-inv N/A

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

       15. lift-/.f64 N/A

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

       16. clear-num N/A

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

       17. lower-*.f64 N/A

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

       18. lift-*.f64 N/A

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

   11. Applied rewrites 84.7%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.85 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+165}:\\ \;\;\;\;\frac{2}{t \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \frac{\ell}{\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot \left(t \cdot \frac{t \cdot k}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3300:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\frac{\ell}{t\_m}}{t\_m \cdot k}\\ \mathbf{elif}\;k \leq 6.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left(t\_m \cdot \left(t\_m \cdot \frac{t\_m \cdot \sin k}{\ell}\right)\right) \cdot \left(2 \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

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 3300.0)
    (* (/ l (* t_m k)) (/ (/ l t_m) (* t_m k)))
    (if (<= k 6.5e+154)
      (/ (* 2.0 l) (* (* t_m (* t_m (/ (* t_m (sin k)) l))) (* 2.0 (tan 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 <= 3300.0) {
		tmp = (l / (t_m * k)) * ((l / t_m) / (t_m * k));
	} else if (k <= 6.5e+154) {
		tmp = (2.0 * l) / ((t_m * (t_m * ((t_m * sin(k)) / l))) * (2.0 * tan(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 <= 3300.0d0) then
        tmp = (l / (t_m * k)) * ((l / t_m) / (t_m * k))
    else if (k <= 6.5d+154) then
        tmp = (2.0d0 * l) / ((t_m * (t_m * ((t_m * sin(k)) / l))) * (2.0d0 * tan(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 <= 3300.0) {
		tmp = (l / (t_m * k)) * ((l / t_m) / (t_m * k));
	} else if (k <= 6.5e+154) {
		tmp = (2.0 * l) / ((t_m * (t_m * ((t_m * Math.sin(k)) / l))) * (2.0 * Math.tan(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 <= 3300.0:
		tmp = (l / (t_m * k)) * ((l / t_m) / (t_m * k))
	elif k <= 6.5e+154:
		tmp = (2.0 * l) / ((t_m * (t_m * ((t_m * math.sin(k)) / l))) * (2.0 * math.tan(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 <= 3300.0)
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(Float64(l / t_m) / Float64(t_m * k)));
	elseif (k <= 6.5e+154)
		tmp = Float64(Float64(2.0 * l) / Float64(Float64(t_m * Float64(t_m * Float64(Float64(t_m * sin(k)) / l))) * Float64(2.0 * tan(k))));
	else
		tmp = Float64(l * Float64(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 <= 3300.0)
		tmp = (l / (t_m * k)) * ((l / t_m) / (t_m * k));
	elseif (k <= 6.5e+154)
		tmp = (2.0 * l) / ((t_m * (t_m * ((t_m * sin(k)) / l))) * (2.0 * tan(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, 3300.0], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 6.5e+154], N[(N[(2.0 * l), $MachinePrecision] / N[(N[(t$95$m * N[(t$95$m * N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(t$95$m * N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3300

   1. Initial program 58.2%

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

   3. Taylor expanded in k around 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 55.7

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

   5. Applied rewrites 55.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. unswap-sqr N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 81.0

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

   7. Applied rewrites 81.0%

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

   if 3300 < k < 6.5000000000000005e154

   1. Initial program 47.2%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 11.6

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

   4. Applied rewrites 11.6%

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

   6. Applied rewrites 62.4%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 66.7%

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

   if 6.5000000000000005e154 < k

   1. Initial program 34.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 34.4

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

   5. Applied rewrites 34.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 35.6

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 35.6

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

   7. Applied rewrites 35.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 43.7

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

   9. Applied rewrites 43.7%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3300:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\frac{\ell}{t}}{t \cdot k}\\ \mathbf{elif}\;k \leq 6.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left(t \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right) \cdot \left(2 \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\frac{\ell}{t\_m}}{t\_m \cdot k}\\ \mathbf{elif}\;k \leq 2.05 \cdot 10^{+145}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(2 \cdot \left(\tan k \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \frac{t\_m}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(k, k \cdot \left(\left(t\_m \cdot \left(k \cdot k\right)\right) \cdot \mathsf{fma}\left(2, \frac{t\_m \cdot t\_m}{\ell} \cdot 0.08611111111111111, \frac{0.16666666666666666}{\ell}\right)\right), 2 \cdot \frac{t\_m \cdot \left(t\_m \cdot t\_m\right)}{\ell}\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2e-24)
    (* (/ l (* t_m k)) (/ (/ l t_m) (* t_m k)))
    (if (<= k 2.05e+145)
      (/ 2.0 (* (sin k) (* 2.0 (* (tan k) (* (* t_m t_m) (/ t_m (* l l)))))))
      (/
       2.0
       (/
        (*
         (* k k)
         (fma
          k
          (*
           k
           (*
            (* t_m (* k k))
            (fma
             2.0
             (* (/ (* t_m t_m) l) 0.08611111111111111)
             (/ 0.16666666666666666 l))))
          (* 2.0 (/ (* t_m (* t_m t_m)) l))))
        l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2e-24) {
		tmp = (l / (t_m * k)) * ((l / t_m) / (t_m * k));
	} else if (k <= 2.05e+145) {
		tmp = 2.0 / (sin(k) * (2.0 * (tan(k) * ((t_m * t_m) * (t_m / (l * l))))));
	} else {
		tmp = 2.0 / (((k * k) * fma(k, (k * ((t_m * (k * k)) * fma(2.0, (((t_m * t_m) / l) * 0.08611111111111111), (0.16666666666666666 / l)))), (2.0 * ((t_m * (t_m * t_m)) / l)))) / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2e-24)
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(Float64(l / t_m) / Float64(t_m * k)));
	elseif (k <= 2.05e+145)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(2.0 * Float64(tan(k) * Float64(Float64(t_m * t_m) * Float64(t_m / Float64(l * l)))))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * fma(k, Float64(k * Float64(Float64(t_m * Float64(k * k)) * fma(2.0, Float64(Float64(Float64(t_m * t_m) / l) * 0.08611111111111111), Float64(0.16666666666666666 / l)))), Float64(2.0 * Float64(Float64(t_m * Float64(t_m * t_m)) / l)))) / l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2e-24], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.05e+145], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(2.0 * N[(N[Tan[k], $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(k * N[(k * N[(N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 0.08611111111111111), $MachinePrecision] + N[(0.16666666666666666 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.99999999999999985e-24

   1. Initial program 58.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. unswap-sqr N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 82.0

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

   7. Applied rewrites 82.0%

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

   if 1.99999999999999985e-24 < k < 2.0500000000000001e145

   1. Initial program 49.3%

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

   4. Applied rewrites 54.3%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 69.6%

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

   if 2.0500000000000001e145 < k

   1. Initial program 34.0%

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

   4. Applied rewrites 20.7%

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

   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. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 49.2%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{\ell} + {k}^{2} \cdot \left(t \cdot \left(2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right) + {k}^{2} \cdot \left(t \cdot \left(2 \cdot \left(\frac{2}{45} \cdot \frac{{t}^{2}}{\ell} - \left(\frac{-1}{2} \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{24} \cdot \frac{{t}^{2}}{\ell}\right)\right) + \frac{1}{6} \cdot \frac{1}{\ell}\right)\right)\right)\right)}}{\ell}} \]

   10. Applied rewrites 18.4%

       \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(k, k \cdot \mathsf{fma}\left(\mathsf{fma}\left(2, \frac{t \cdot t}{\ell} \cdot 0.044444444444444446 - \mathsf{fma}\left(\frac{t \cdot t}{\ell}, 0.041666666666666664, \frac{t \cdot t}{\ell} \cdot -0.08333333333333333\right), \frac{0.16666666666666666}{\ell}\right), \left(k \cdot k\right) \cdot t, t \cdot \mathsf{fma}\left(\frac{t \cdot t}{\ell}, 0.3333333333333333, \frac{1}{\ell}\right)\right), 2 \cdot \frac{t \cdot \left(t \cdot t\right)}{\ell}\right)}}{\ell}} \]

   11. Taylor expanded in k around inf

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(k, k \cdot \color{blue}{\left({k}^{2} \cdot \left(t \cdot \left(2 \cdot \left(\frac{2}{45} \cdot \frac{{t}^{2}}{\ell} - \left(\frac{-1}{12} \cdot \frac{{t}^{2}}{\ell} + \frac{1}{24} \cdot \frac{{t}^{2}}{\ell}\right)\right) + \frac{1}{6} \cdot \frac{1}{\ell}\right)\right)\right)}, 2 \cdot \frac{t \cdot \left(t \cdot t\right)}{\ell}\right)}{\ell}} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(k, k \cdot \color{blue}{\left(\left({k}^{2} \cdot t\right) \cdot \left(2 \cdot \left(\frac{2}{45} \cdot \frac{{t}^{2}}{\ell} - \left(\frac{-1}{12} \cdot \frac{{t}^{2}}{\ell} + \frac{1}{24} \cdot \frac{{t}^{2}}{\ell}\right)\right) + \frac{1}{6} \cdot \frac{1}{\ell}\right)\right)}, 2 \cdot \frac{t \cdot \left(t \cdot t\right)}{\ell}\right)}{\ell}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(k, k \cdot \color{blue}{\left(\left({k}^{2} \cdot t\right) \cdot \left(2 \cdot \left(\frac{2}{45} \cdot \frac{{t}^{2}}{\ell} - \left(\frac{-1}{12} \cdot \frac{{t}^{2}}{\ell} + \frac{1}{24} \cdot \frac{{t}^{2}}{\ell}\right)\right) + \frac{1}{6} \cdot \frac{1}{\ell}\right)\right)}, 2 \cdot \frac{t \cdot \left(t \cdot t\right)}{\ell}\right)}{\ell}} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(k, k \cdot \left(\color{blue}{\left({k}^{2} \cdot t\right)} \cdot \left(2 \cdot \left(\frac{2}{45} \cdot \frac{{t}^{2}}{\ell} - \left(\frac{-1}{12} \cdot \frac{{t}^{2}}{\ell} + \frac{1}{24} \cdot \frac{{t}^{2}}{\ell}\right)\right) + \frac{1}{6} \cdot \frac{1}{\ell}\right)\right), 2 \cdot \frac{t \cdot \left(t \cdot t\right)}{\ell}\right)}{\ell}} \]

       4. unpow2 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(k, k \cdot \left(\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot \left(2 \cdot \left(\frac{2}{45} \cdot \frac{{t}^{2}}{\ell} - \left(\frac{-1}{12} \cdot \frac{{t}^{2}}{\ell} + \frac{1}{24} \cdot \frac{{t}^{2}}{\ell}\right)\right) + \frac{1}{6} \cdot \frac{1}{\ell}\right)\right), 2 \cdot \frac{t \cdot \left(t \cdot t\right)}{\ell}\right)}{\ell}} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(k, k \cdot \left(\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot \left(2 \cdot \left(\frac{2}{45} \cdot \frac{{t}^{2}}{\ell} - \left(\frac{-1}{12} \cdot \frac{{t}^{2}}{\ell} + \frac{1}{24} \cdot \frac{{t}^{2}}{\ell}\right)\right) + \frac{1}{6} \cdot \frac{1}{\ell}\right)\right), 2 \cdot \frac{t \cdot \left(t \cdot t\right)}{\ell}\right)}{\ell}} \]

       6. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(k, k \cdot \left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\mathsf{fma}\left(2, \frac{2}{45} \cdot \frac{{t}^{2}}{\ell} - \left(\frac{-1}{12} \cdot \frac{{t}^{2}}{\ell} + \frac{1}{24} \cdot \frac{{t}^{2}}{\ell}\right), \frac{1}{6} \cdot \frac{1}{\ell}\right)}\right), 2 \cdot \frac{t \cdot \left(t \cdot t\right)}{\ell}\right)}{\ell}} \]

   13. Applied rewrites 42.9%

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(k, k \cdot \color{blue}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot t}{\ell} \cdot 0.08611111111111111, \frac{0.16666666666666666}{\ell}\right)\right)}, 2 \cdot \frac{t \cdot \left(t \cdot t\right)}{\ell}\right)}{\ell}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\frac{\ell}{t}}{t \cdot k}\\ \mathbf{elif}\;k \leq 2.05 \cdot 10^{+145}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(2 \cdot \left(\tan k \cdot \left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(k, k \cdot \left(\left(t \cdot \left(k \cdot k\right)\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot t}{\ell} \cdot 0.08611111111111111, \frac{0.16666666666666666}{\ell}\right)\right), 2 \cdot \frac{t \cdot \left(t \cdot t\right)}{\ell}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{-26}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\frac{\ell}{t\_m}}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2}{t\_m \cdot \left(\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.9e-26)
    (* (/ l (* t_m k)) (/ (/ l t_m) (* t_m k)))
    (*
     l
     (/
      2.0
      (* t_m (* (* (/ (sin k) l) (tan k)) (fma 2.0 (* 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 <= 1.9e-26) {
		tmp = (l / (t_m * k)) * ((l / t_m) / (t_m * k));
	} else {
		tmp = l * (2.0 / (t_m * (((sin(k) / l) * tan(k)) * fma(2.0, (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 <= 1.9e-26)
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(Float64(l / t_m) / Float64(t_m * k)));
	else
		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))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.9e-26], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.90000000000000007e-26

   1. Initial program 58.9%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-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 55.8

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

   5. Applied rewrites 55.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. unswap-sqr N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 81.9

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

   7. Applied rewrites 81.9%

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

   if 1.90000000000000007e-26 < k

   1. Initial program 42.2%

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

   4. Applied rewrites 43.4%

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

   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. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 76.6%

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

   9. Applied rewrites 83.3%

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

   11. Applied rewrites 77.0%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{-26}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\frac{\ell}{t}}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2}{t \cdot \left(\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot t\_m}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{-72}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, t\_2, \left(k \cdot k\right) \cdot \mathsf{fma}\left(t\_2, 0.3333333333333333, \frac{1}{\ell}\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\frac{\ell}{t\_m}}{t\_m \cdot k}\\ \end{array} \end{array} \end{array} \]

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 8e-72)
      (/
       2.0
       (/
        (*
         t_m
         (*
          (* k k)
          (fma 2.0 t_2 (* (* k k) (fma t_2 0.3333333333333333 (/ 1.0 l))))))
        l))
      (* (/ l (* t_m k)) (/ (/ l t_m) (* t_m k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m * t_m) / l;
	double tmp;
	if (t_m <= 8e-72) {
		tmp = 2.0 / ((t_m * ((k * k) * fma(2.0, t_2, ((k * k) * fma(t_2, 0.3333333333333333, (1.0 / l)))))) / l);
	} else {
		tmp = (l / (t_m * k)) * ((l / t_m) / (t_m * k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(t_m * t_m) / l)
	tmp = 0.0
	if (t_m <= 8e-72)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(k * k) * fma(2.0, t_2, Float64(Float64(k * k) * fma(t_2, 0.3333333333333333, Float64(1.0 / l)))))) / l));
	else
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(Float64(l / t_m) / Float64(t_m * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 8e-72], N[(2.0 / N[(N[(t$95$m * N[(N[(k * k), $MachinePrecision] * N[(2.0 * t$95$2 + N[(N[(k * k), $MachinePrecision] * N[(t$95$2 * 0.3333333333333333 + N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.9999999999999997e-72

   1. Initial program 54.4%

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

   4. Applied rewrites 45.6%

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

   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. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 81.2%

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

   8. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

   10. Applied rewrites 69.0%

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

   if 7.9999999999999997e-72 < t

   1. Initial program 54.0%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 43.9

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

   5. Applied rewrites 43.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. unswap-sqr N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 76.2

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

   7. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot k} \cdot \frac{\frac{\ell}{t}}{t \cdot k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 70.8% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.8 \cdot 10^{-72}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{elif}\;t\_m \leq 3.2 \cdot 10^{+116}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\ell}{k \cdot \left(t\_m \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 5.8e-72)
    (* l (/ l (* t_m (* t_m (* t_m (* k k))))))
    (if (<= t_m 3.2e+116)
      (* (/ l (* t_m k)) (/ l (* k (* t_m t_m))))
      (* l (/ l (* (* t_m k) (* t_m (* t_m k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.8e-72) {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	} else if (t_m <= 3.2e+116) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else {
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 5.8d-72) then
        tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))))
    else if (t_m <= 3.2d+116) then
        tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)))
    else
        tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.8e-72) {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	} else if (t_m <= 3.2e+116) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else {
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 5.8e-72:
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))))
	elif t_m <= 3.2e+116:
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)))
	else:
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.8e-72)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k))))));
	elseif (t_m <= 3.2e+116)
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(l / Float64(k * Float64(t_m * t_m))));
	else
		tmp = Float64(l * Float64(l / Float64(Float64(t_m * k) * Float64(t_m * Float64(t_m * k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 5.8e-72)
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	elseif (t_m <= 3.2e+116)
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	else
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.8e-72], N[(l * N[(l / N[(t$95$m * N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.2e+116], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.79999999999999995e-72

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 56.3

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

   5. Applied rewrites 56.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 60.8

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 62.5

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

   7. Applied rewrites 62.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 66.1

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

   9. Applied rewrites 66.1%

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

   if 5.79999999999999995e-72 < t < 3.2e116

   1. Initial program 62.9%

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

   3. 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 50.4

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

   5. Applied rewrites 50.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 53.4

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 53.4

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

   7. Applied rewrites 53.4%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 61.3

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

   9. Applied rewrites 61.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l/ N/A

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

       6. lift-*.f64 N/A

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

       7. times-frac N/A

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

       8. lift-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*r* N/A

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 71.6

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

       17. lift-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lower-*.f64 71.6

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

   11. Applied rewrites 71.6%

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

   if 3.2e116 < t

   1. Initial program 46.2%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 38.2

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

   5. Applied rewrites 38.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 41.8

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      12. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      14. lower-*.f64 46.6

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. Applied rewrites 46.6%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k}} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot k} \cdot \ell \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right)} \cdot k} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\color{blue}{\left(\left(t \cdot t\right) \cdot k\right)} \cdot t\right) \cdot k} \cdot \ell \]

      7. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot k} \cdot \ell \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right)} \cdot k} \cdot \ell \]

      9. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\color{blue}{\left(t \cdot t\right)} \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      13. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      16. lower-*.f64 72.6

          \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

   9. Applied rewrites 72.6%

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{-72}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+116}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\ell}{k \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.1% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-24}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\frac{\ell}{t\_m}}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.1e-24)
    (* (/ l (* t_m k)) (/ (/ l t_m) (* t_m k)))
    (/ (/ (* l l) (* t_m (* t_m (* k k)))) t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.1e-24) {
		tmp = (l / (t_m * k)) * ((l / t_m) / (t_m * k));
	} else {
		tmp = ((l * l) / (t_m * (t_m * (k * k)))) / t_m;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.1d-24) then
        tmp = (l / (t_m * k)) * ((l / t_m) / (t_m * k))
    else
        tmp = ((l * l) / (t_m * (t_m * (k * k)))) / t_m
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.1e-24) {
		tmp = (l / (t_m * k)) * ((l / t_m) / (t_m * k));
	} else {
		tmp = ((l * l) / (t_m * (t_m * (k * k)))) / t_m;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 2.1e-24:
		tmp = (l / (t_m * k)) * ((l / t_m) / (t_m * k))
	else:
		tmp = ((l * l) / (t_m * (t_m * (k * k)))) / t_m
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.1e-24)
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(Float64(l / t_m) / Float64(t_m * k)));
	else
		tmp = Float64(Float64(Float64(l * l) / Float64(t_m * Float64(t_m * Float64(k * k)))) / t_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 2.1e-24)
		tmp = (l / (t_m * k)) * ((l / t_m) / (t_m * k));
	else
		tmp = ((l * l) / (t_m * (t_m * (k * k)))) / t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.1e-24], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.0999999999999999e-24

   1. Initial program 58.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \]

      7. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \frac{\ell}{t}}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \frac{\ell}{t}}{\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \frac{\ell}{t}}{\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      10. unswap-sqr N/A

          \[\leadsto \frac{\ell \cdot \frac{\ell}{t}}{\color{blue}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}} \]

      11. times-frac N/A

          \[\leadsto \color{blue}{\frac{\ell}{t \cdot k} \cdot \frac{\frac{\ell}{t}}{t \cdot k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{t \cdot k} \cdot \frac{\frac{\ell}{t}}{t \cdot k}} \]

      13. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{t \cdot k}} \cdot \frac{\frac{\ell}{t}}{t \cdot k} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot k}} \cdot \frac{\frac{\ell}{t}}{t \cdot k} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot k} \cdot \color{blue}{\frac{\frac{\ell}{t}}{t \cdot k}} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot k} \cdot \frac{\color{blue}{\frac{\ell}{t}}}{t \cdot k} \]

      17. lower-*.f64 82.0

          \[\leadsto \frac{\ell}{t \cdot k} \cdot \frac{\frac{\ell}{t}}{\color{blue}{t \cdot k}} \]

   7. Applied rewrites 82.0%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot k} \cdot \frac{\frac{\ell}{t}}{t \cdot k}} \]

   if 2.0999999999999999e-24 < k

   1. Initial program 42.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 44.6

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 44.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 44.5

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      12. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      14. lower-*.f64 44.5

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. Applied rewrites 44.5%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k}} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot k} \cdot \ell \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right)} \cdot k} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\color{blue}{\left(\left(t \cdot t\right) \cdot k\right)} \cdot t\right) \cdot k} \cdot \ell \]

      7. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot k} \cdot \ell \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right)} \cdot k} \cdot \ell \]

      9. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\color{blue}{\left(t \cdot t\right)} \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      13. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      16. lower-*.f64 43.5

          \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

   9. Applied rewrites 43.5%

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

       4. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{t \cdot \left(t \cdot k\right)}}{t \cdot k}} \cdot \ell \]

       5. lift-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}{t \cdot k} \cdot \ell \]

       6. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{t \cdot \left(t \cdot k\right)} \cdot \ell}{t \cdot k}} \]

       7. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\ell \cdot \frac{\ell}{t \cdot \left(t \cdot k\right)}}}{t \cdot k} \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \frac{\ell}{t \cdot \left(t \cdot k\right)}}{\color{blue}{t \cdot k}} \]

       9. frac-times N/A

          \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\frac{\ell}{t \cdot \left(t \cdot k\right)}}{k}} \]

       10. lift-/.f64 N/A

           \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\frac{\ell}{t \cdot \left(t \cdot k\right)}}{k}} \]

       11. associate-*l/ N/A

           \[\leadsto \color{blue}{\frac{\ell \cdot \frac{\frac{\ell}{t \cdot \left(t \cdot k\right)}}{k}}{t}} \]

       12. lower-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\ell \cdot \frac{\frac{\ell}{t \cdot \left(t \cdot k\right)}}{k}}{t}} \]

   11. Applied rewrites 50.5%

       \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 71.0% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot \frac{t\_m \cdot k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\frac{\ell}{t\_m}}{t\_m \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 5e-136)
    (/ l (* t_m (* k (* t_m (/ (* t_m k) l)))))
    (* (/ 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 <= 5e-136) {
		tmp = l / (t_m * (k * (t_m * ((t_m * k) / l))));
	} 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 <= 5d-136) then
        tmp = l / (t_m * (k * (t_m * ((t_m * k) / l))))
    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 <= 5e-136) {
		tmp = l / (t_m * (k * (t_m * ((t_m * k) / l))));
	} 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 <= 5e-136:
		tmp = l / (t_m * (k * (t_m * ((t_m * k) / l))))
	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 <= 5e-136)
		tmp = Float64(l / Float64(t_m * Float64(k * Float64(t_m * Float64(Float64(t_m * k) / l)))));
	else
		tmp = Float64(Float64(l / t_m) * Float64(Float64(l / 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 <= 5e-136)
		tmp = l / (t_m * (k * (t_m * ((t_m * k) / l))));
	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, 5e-136], N[(l / N[(t$95$m * N[(k * N[(t$95$m * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.0000000000000002e-136

   1. Initial program 56.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-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 51.7

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 57.1

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      12. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      14. lower-*.f64 60.3

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. Applied rewrites 60.3%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k}} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot k} \cdot \ell \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right)} \cdot k} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\color{blue}{\left(\left(t \cdot t\right) \cdot k\right)} \cdot t\right) \cdot k} \cdot \ell \]

      7. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot k} \cdot \ell \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right)} \cdot k} \cdot \ell \]

      9. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\color{blue}{\left(t \cdot t\right)} \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      13. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      16. lower-*.f64 74.4

          \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

   9. Applied rewrites 74.4%

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

       4. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{t \cdot \left(t \cdot k\right)}}{t \cdot k}} \cdot \ell \]

       5. lift-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}{t \cdot k} \cdot \ell \]

       6. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{t \cdot \left(t \cdot k\right)} \cdot \ell}{t \cdot k}} \]

       7. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\ell \cdot \frac{\ell}{t \cdot \left(t \cdot k\right)}}}{t \cdot k} \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \frac{\ell}{t \cdot \left(t \cdot k\right)}}{\color{blue}{t \cdot k}} \]

       9. frac-times N/A

          \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\frac{\ell}{t \cdot \left(t \cdot k\right)}}{k}} \]

       10. clear-num N/A

           \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{1}{\frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}} \]

       11. frac-times N/A

           \[\leadsto \color{blue}{\frac{\ell \cdot 1}{t \cdot \frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}} \]

       12. metadata-eval N/A

           \[\leadsto \frac{\ell \cdot \color{blue}{\frac{1}{1}}}{t \cdot \frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}} \]

       13. div-inv N/A

           \[\leadsto \frac{\color{blue}{\frac{\ell}{1}}}{t \cdot \frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}} \]

       14. /-rgt-identity N/A

           \[\leadsto \frac{\color{blue}{\ell}}{t \cdot \frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}} \]

       15. lower-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\ell}{t \cdot \frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}} \]

       16. lower-*.f64 N/A

           \[\leadsto \frac{\ell}{\color{blue}{t \cdot \frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}} \]

       17. div-inv N/A

           \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(k \cdot \frac{1}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}\right)}} \]

       18. lift-/.f64 N/A

           \[\leadsto \frac{\ell}{t \cdot \left(k \cdot \frac{1}{\color{blue}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}\right)} \]

   11. Applied rewrites 79.5%

       \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \frac{k \cdot t}{\ell}\right)\right)}} \]

   if 5.0000000000000002e-136 < k

   1. Initial program 50.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \]

      7. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \frac{\ell}{t}}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \frac{\ell}{t}}{\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \frac{\ell}{t}}{\color{blue}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      10. times-frac N/A

          \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\frac{\ell}{t}}{t \cdot \left(k \cdot k\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\frac{\ell}{t}}{t \cdot \left(k \cdot k\right)}} \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{t}} \cdot \frac{\frac{\ell}{t}}{t \cdot \left(k \cdot k\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot k\right)}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{t} \cdot \frac{\color{blue}{\frac{\ell}{t}}}{t \cdot \left(k \cdot k\right)} \]

      15. lower-*.f64 61.0

          \[\leadsto \frac{\ell}{t} \cdot \frac{\frac{\ell}{t}}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]

   7. Applied rewrites 61.0%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\frac{\ell}{t}}{t \cdot \left(k \cdot k\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \frac{t \cdot k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{\ell}{t}}{t \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 70.5% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot \frac{t\_m \cdot k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2e-24)
    (/ l (* t_m (* k (* t_m (/ (* t_m k) l)))))
    (/ (/ (* l l) (* t_m (* t_m (* k k)))) t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2e-24) {
		tmp = l / (t_m * (k * (t_m * ((t_m * k) / l))));
	} else {
		tmp = ((l * l) / (t_m * (t_m * (k * k)))) / t_m;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2d-24) then
        tmp = l / (t_m * (k * (t_m * ((t_m * k) / l))))
    else
        tmp = ((l * l) / (t_m * (t_m * (k * k)))) / t_m
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2e-24) {
		tmp = l / (t_m * (k * (t_m * ((t_m * k) / l))));
	} else {
		tmp = ((l * l) / (t_m * (t_m * (k * k)))) / t_m;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 2e-24:
		tmp = l / (t_m * (k * (t_m * ((t_m * k) / l))))
	else:
		tmp = ((l * l) / (t_m * (t_m * (k * k)))) / t_m
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2e-24)
		tmp = Float64(l / Float64(t_m * Float64(k * Float64(t_m * Float64(Float64(t_m * k) / l)))));
	else
		tmp = Float64(Float64(Float64(l * l) / Float64(t_m * Float64(t_m * Float64(k * k)))) / t_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 2e-24)
		tmp = l / (t_m * (k * (t_m * ((t_m * k) / l))));
	else
		tmp = ((l * l) / (t_m * (t_m * (k * k)))) / t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2e-24], N[(l / N[(t$95$m * N[(k * N[(t$95$m * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.99999999999999985e-24

   1. Initial program 58.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 61.2

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      12. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      14. lower-*.f64 63.9

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k}} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot k} \cdot \ell \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right)} \cdot k} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\color{blue}{\left(\left(t \cdot t\right) \cdot k\right)} \cdot t\right) \cdot k} \cdot \ell \]

      7. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot k} \cdot \ell \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right)} \cdot k} \cdot \ell \]

      9. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\color{blue}{\left(t \cdot t\right)} \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      13. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      16. lower-*.f64 75.8

          \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

   9. Applied rewrites 75.8%

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

       4. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{t \cdot \left(t \cdot k\right)}}{t \cdot k}} \cdot \ell \]

       5. lift-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}{t \cdot k} \cdot \ell \]

       6. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{t \cdot \left(t \cdot k\right)} \cdot \ell}{t \cdot k}} \]

       7. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\ell \cdot \frac{\ell}{t \cdot \left(t \cdot k\right)}}}{t \cdot k} \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \frac{\ell}{t \cdot \left(t \cdot k\right)}}{\color{blue}{t \cdot k}} \]

       9. frac-times N/A

          \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\frac{\ell}{t \cdot \left(t \cdot k\right)}}{k}} \]

       10. clear-num N/A

           \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{1}{\frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}} \]

       11. frac-times N/A

           \[\leadsto \color{blue}{\frac{\ell \cdot 1}{t \cdot \frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}} \]

       12. metadata-eval N/A

           \[\leadsto \frac{\ell \cdot \color{blue}{\frac{1}{1}}}{t \cdot \frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}} \]

       13. div-inv N/A

           \[\leadsto \frac{\color{blue}{\frac{\ell}{1}}}{t \cdot \frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}} \]

       14. /-rgt-identity N/A

           \[\leadsto \frac{\color{blue}{\ell}}{t \cdot \frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}} \]

       15. lower-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\ell}{t \cdot \frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}} \]

       16. lower-*.f64 N/A

           \[\leadsto \frac{\ell}{\color{blue}{t \cdot \frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}} \]

       17. div-inv N/A

           \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(k \cdot \frac{1}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}\right)}} \]

       18. lift-/.f64 N/A

           \[\leadsto \frac{\ell}{t \cdot \left(k \cdot \frac{1}{\color{blue}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}\right)} \]

   11. Applied rewrites 80.6%

       \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \frac{k \cdot t}{\ell}\right)\right)}} \]

   if 1.99999999999999985e-24 < k

   1. Initial program 42.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 44.6

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 44.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 44.5

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      12. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      14. lower-*.f64 44.5

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. Applied rewrites 44.5%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k}} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot k} \cdot \ell \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right)} \cdot k} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\color{blue}{\left(\left(t \cdot t\right) \cdot k\right)} \cdot t\right) \cdot k} \cdot \ell \]

      7. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot k} \cdot \ell \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right)} \cdot k} \cdot \ell \]

      9. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\color{blue}{\left(t \cdot t\right)} \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      13. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      16. lower-*.f64 43.5

          \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

   9. Applied rewrites 43.5%

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

       4. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{t \cdot \left(t \cdot k\right)}}{t \cdot k}} \cdot \ell \]

       5. lift-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}{t \cdot k} \cdot \ell \]

       6. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{t \cdot \left(t \cdot k\right)} \cdot \ell}{t \cdot k}} \]

       7. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\ell \cdot \frac{\ell}{t \cdot \left(t \cdot k\right)}}}{t \cdot k} \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \frac{\ell}{t \cdot \left(t \cdot k\right)}}{\color{blue}{t \cdot k}} \]

       9. frac-times N/A

          \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\frac{\ell}{t \cdot \left(t \cdot k\right)}}{k}} \]

       10. lift-/.f64 N/A

           \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\frac{\ell}{t \cdot \left(t \cdot k\right)}}{k}} \]

       11. associate-*l/ N/A

           \[\leadsto \color{blue}{\frac{\ell \cdot \frac{\frac{\ell}{t \cdot \left(t \cdot k\right)}}{k}}{t}} \]

       12. lower-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\ell \cdot \frac{\frac{\ell}{t \cdot \left(t \cdot k\right)}}{k}}{t}} \]

   11. Applied rewrites 50.5%

       \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \frac{t \cdot k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 70.5% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot \frac{t\_m \cdot k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t\_m}}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 1.5e-24)
    (/ l (* t_m (* k (* t_m (/ (* t_m k) l)))))
    (/ (/ (* 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 <= 1.5e-24) {
		tmp = l / (t_m * (k * (t_m * ((t_m * k) / l))));
	} 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 <= 1.5d-24) then
        tmp = l / (t_m * (k * (t_m * ((t_m * k) / l))))
    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 <= 1.5e-24) {
		tmp = l / (t_m * (k * (t_m * ((t_m * k) / l))));
	} 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 <= 1.5e-24:
		tmp = l / (t_m * (k * (t_m * ((t_m * k) / l))))
	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 <= 1.5e-24)
		tmp = Float64(l / Float64(t_m * Float64(k * Float64(t_m * Float64(Float64(t_m * k) / l)))));
	else
		tmp = Float64(Float64(Float64(l * l) / 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 <= 1.5e-24)
		tmp = l / (t_m * (k * (t_m * ((t_m * k) / l))));
	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, 1.5e-24], N[(l / N[(t$95$m * N[(k * N[(t$95$m * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.49999999999999998e-24

   1. Initial program 58.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 61.2

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      12. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      14. lower-*.f64 63.9

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k}} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot k} \cdot \ell \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right)} \cdot k} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\color{blue}{\left(\left(t \cdot t\right) \cdot k\right)} \cdot t\right) \cdot k} \cdot \ell \]

      7. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot k} \cdot \ell \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right)} \cdot k} \cdot \ell \]

      9. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\color{blue}{\left(t \cdot t\right)} \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      13. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      16. lower-*.f64 75.8

          \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

   9. Applied rewrites 75.8%

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

       4. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{t \cdot \left(t \cdot k\right)}}{t \cdot k}} \cdot \ell \]

       5. lift-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}{t \cdot k} \cdot \ell \]

       6. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{t \cdot \left(t \cdot k\right)} \cdot \ell}{t \cdot k}} \]

       7. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\ell \cdot \frac{\ell}{t \cdot \left(t \cdot k\right)}}}{t \cdot k} \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \frac{\ell}{t \cdot \left(t \cdot k\right)}}{\color{blue}{t \cdot k}} \]

       9. frac-times N/A

          \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\frac{\ell}{t \cdot \left(t \cdot k\right)}}{k}} \]

       10. clear-num N/A

           \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{1}{\frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}} \]

       11. frac-times N/A

           \[\leadsto \color{blue}{\frac{\ell \cdot 1}{t \cdot \frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}} \]

       12. metadata-eval N/A

           \[\leadsto \frac{\ell \cdot \color{blue}{\frac{1}{1}}}{t \cdot \frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}} \]

       13. div-inv N/A

           \[\leadsto \frac{\color{blue}{\frac{\ell}{1}}}{t \cdot \frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}} \]

       14. /-rgt-identity N/A

           \[\leadsto \frac{\color{blue}{\ell}}{t \cdot \frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}} \]

       15. lower-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\ell}{t \cdot \frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}} \]

       16. lower-*.f64 N/A

           \[\leadsto \frac{\ell}{\color{blue}{t \cdot \frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}} \]

       17. div-inv N/A

           \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(k \cdot \frac{1}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}\right)}} \]

       18. lift-/.f64 N/A

           \[\leadsto \frac{\ell}{t \cdot \left(k \cdot \frac{1}{\color{blue}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}\right)} \]

   11. Applied rewrites 80.6%

       \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \frac{k \cdot t}{\ell}\right)\right)}} \]

   if 1.49999999999999998e-24 < k

   1. Initial program 42.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 44.6

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 44.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\ell \cdot \ell}{t}}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{\color{blue}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{\color{blue}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      11. lower-*.f64 50.4

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \]

   7. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \frac{t \cdot k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t}}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 70.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+97}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot \frac{t\_m \cdot k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

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 2e+97)
    (/ l (* t_m (* k (* t_m (/ (* t_m k) l)))))
    (/ (* 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 <= 2e+97) {
		tmp = l / (t_m * (k * (t_m * ((t_m * k) / l))));
	} 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 <= 2d+97) then
        tmp = l / (t_m * (k * (t_m * ((t_m * k) / l))))
    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 <= 2e+97) {
		tmp = l / (t_m * (k * (t_m * ((t_m * k) / l))));
	} 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 <= 2e+97:
		tmp = l / (t_m * (k * (t_m * ((t_m * k) / l))))
	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 <= 2e+97)
		tmp = Float64(l / Float64(t_m * Float64(k * Float64(t_m * Float64(Float64(t_m * k) / l)))));
	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 <= 2e+97)
		tmp = l / (t_m * (k * (t_m * ((t_m * k) / l))));
	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, 2e+97], N[(l / N[(t$95$m * N[(k * N[(t$95$m * N[(N[(t$95$m * k), $MachinePrecision] / l), $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 < 2.0000000000000001e97

   1. Initial program 56.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 54.6

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 59.3

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      12. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      14. lower-*.f64 61.6

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. Applied rewrites 61.6%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k}} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot k} \cdot \ell \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right)} \cdot k} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\color{blue}{\left(\left(t \cdot t\right) \cdot k\right)} \cdot t\right) \cdot k} \cdot \ell \]

      7. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot k} \cdot \ell \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right)} \cdot k} \cdot \ell \]

      9. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\color{blue}{\left(t \cdot t\right)} \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      13. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      16. lower-*.f64 71.7

          \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

   9. Applied rewrites 71.7%

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

       4. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{t \cdot \left(t \cdot k\right)}}{t \cdot k}} \cdot \ell \]

       5. lift-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}{t \cdot k} \cdot \ell \]

       6. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{t \cdot \left(t \cdot k\right)} \cdot \ell}{t \cdot k}} \]

       7. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\ell \cdot \frac{\ell}{t \cdot \left(t \cdot k\right)}}}{t \cdot k} \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \frac{\ell}{t \cdot \left(t \cdot k\right)}}{\color{blue}{t \cdot k}} \]

       9. frac-times N/A

          \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\frac{\ell}{t \cdot \left(t \cdot k\right)}}{k}} \]

       10. clear-num N/A

           \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{1}{\frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}} \]

       11. frac-times N/A

           \[\leadsto \color{blue}{\frac{\ell \cdot 1}{t \cdot \frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}} \]

       12. metadata-eval N/A

           \[\leadsto \frac{\ell \cdot \color{blue}{\frac{1}{1}}}{t \cdot \frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}} \]

       13. div-inv N/A

           \[\leadsto \frac{\color{blue}{\frac{\ell}{1}}}{t \cdot \frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}} \]

       14. /-rgt-identity N/A

           \[\leadsto \frac{\color{blue}{\ell}}{t \cdot \frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}} \]

       15. lower-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\ell}{t \cdot \frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}} \]

       16. lower-*.f64 N/A

           \[\leadsto \frac{\ell}{\color{blue}{t \cdot \frac{k}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}} \]

       17. div-inv N/A

           \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(k \cdot \frac{1}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}\right)}} \]

       18. lift-/.f64 N/A

           \[\leadsto \frac{\ell}{t \cdot \left(k \cdot \frac{1}{\color{blue}{\frac{\ell}{t \cdot \left(t \cdot k\right)}}}\right)} \]

   11. Applied rewrites 76.3%

       \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \frac{k \cdot t}{\ell}\right)\right)}} \]

   if 2.0000000000000001e97 < k

   1. Initial program 40.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-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 40.4

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 40.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)\right) \cdot t} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot t} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot t} \]

      9. lower-*.f64 48.6

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right) \cdot t} \]

   7. Applied rewrites 48.6%

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right) \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+97}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \frac{t \cdot k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 68.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-156}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 4.2e-156)
    (* l (/ l (* (* t_m k) (* t_m (* 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 <= 4.2e-156) {
		tmp = l * (l / ((t_m * k) * (t_m * (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 <= 4.2d-156) then
        tmp = l * (l / ((t_m * k) * (t_m * (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 <= 4.2e-156) {
		tmp = l * (l / ((t_m * k) * (t_m * (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 <= 4.2e-156:
		tmp = l * (l / ((t_m * k) * (t_m * (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 <= 4.2e-156)
		tmp = Float64(l * Float64(l / Float64(Float64(t_m * k) * Float64(t_m * 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 <= 4.2e-156)
		tmp = l * (l / ((t_m * k) * (t_m * (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, 4.2e-156], N[(l * N[(l / N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * 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 < 4.20000000000000025e-156

   1. Initial program 56.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 50.4

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 54.8

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      12. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      14. lower-*.f64 58.2

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. Applied rewrites 58.2%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k}} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot k} \cdot \ell \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right)} \cdot k} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\color{blue}{\left(\left(t \cdot t\right) \cdot k\right)} \cdot t\right) \cdot k} \cdot \ell \]

      7. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot k} \cdot \ell \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right)} \cdot k} \cdot \ell \]

      9. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\color{blue}{\left(t \cdot t\right)} \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      13. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      16. lower-*.f64 73.0

          \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

   9. Applied rewrites 73.0%

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

   if 4.20000000000000025e-156 < k

   1. Initial program 51.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 55.4

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 55.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{t}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \cdot \frac{\ell}{t} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{t} \]

      11. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \frac{\ell}{t} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \frac{\ell}{t} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \frac{\ell}{t} \]

      14. lower-/.f64 62.9

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

   7. Applied rewrites 62.9%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \frac{\ell}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-156}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 67.2% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-24}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

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 1.5e-24)
    (* l (/ l (* (* t_m k) (* t_m (* 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 <= 1.5e-24) {
		tmp = l * (l / ((t_m * k) * (t_m * (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 <= 1.5d-24) then
        tmp = l * (l / ((t_m * k) * (t_m * (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 <= 1.5e-24) {
		tmp = l * (l / ((t_m * k) * (t_m * (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 <= 1.5e-24:
		tmp = l * (l / ((t_m * k) * (t_m * (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 <= 1.5e-24)
		tmp = Float64(l * Float64(l / Float64(Float64(t_m * k) * Float64(t_m * 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 <= 1.5e-24)
		tmp = l * (l / ((t_m * k) * (t_m * (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, 1.5e-24], N[(l * N[(l / N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * 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 < 1.49999999999999998e-24

   1. Initial program 58.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 61.2

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      12. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      14. lower-*.f64 63.9

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k}} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot k} \cdot \ell \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right)} \cdot k} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\color{blue}{\left(\left(t \cdot t\right) \cdot k\right)} \cdot t\right) \cdot k} \cdot \ell \]

      7. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot k} \cdot \ell \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right)} \cdot k} \cdot \ell \]

      9. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\color{blue}{\left(t \cdot t\right)} \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      13. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      16. lower-*.f64 75.8

          \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

   9. Applied rewrites 75.8%

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

   if 1.49999999999999998e-24 < k

   1. Initial program 42.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 44.6

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 44.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)\right) \cdot t} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot t} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot t} \]

      9. lower-*.f64 49.0

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right) \cdot t} \]

   7. Applied rewrites 49.0%

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right) \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-24}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 67.7% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10^{-132}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

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 1e-132)
    (* l (/ l (* (* t_m k) (* t_m (* 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 <= 1e-132) {
		tmp = l * (l / ((t_m * k) * (t_m * (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 <= 1d-132) then
        tmp = l * (l / ((t_m * k) * (t_m * (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 <= 1e-132) {
		tmp = l * (l / ((t_m * k) * (t_m * (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 <= 1e-132:
		tmp = l * (l / ((t_m * k) * (t_m * (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 <= 1e-132)
		tmp = Float64(l * Float64(l / Float64(Float64(t_m * k) * Float64(t_m * Float64(t_m * k)))));
	else
		tmp = Float64(l * Float64(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 <= 1e-132)
		tmp = l * (l / ((t_m * k) * (t_m * (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, 1e-132], N[(l * N[(l / N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(t$95$m * N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.9999999999999999e-133

   1. Initial program 56.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 51.7

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 57.0

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      12. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      14. lower-*.f64 60.2

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k}} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot k} \cdot \ell \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right)} \cdot k} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\color{blue}{\left(\left(t \cdot t\right) \cdot k\right)} \cdot t\right) \cdot k} \cdot \ell \]

      7. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot k} \cdot \ell \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right)} \cdot k} \cdot \ell \]

      9. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\color{blue}{\left(t \cdot t\right)} \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      13. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      16. lower-*.f64 74.1

          \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

   9. Applied rewrites 74.1%

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

   if 9.9999999999999999e-133 < k

   1. Initial program 50.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. 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 53.8

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 53.8%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 55.9

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      12. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      14. lower-*.f64 55.9

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

      3. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \cdot \ell \]

      4. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \cdot \ell \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)\right) \cdot t} \cdot \ell \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot t} \cdot \ell \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right) \cdot t} \cdot \ell \]

      9. lower-*.f64 58.3

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot t} \cdot \ell \]

   9. Applied rewrites 58.3%

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right) \cdot t}} \cdot \ell \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-132}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 66.7% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}\right) \end{array} \]

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 k) (* t_m (* t_m k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / ((t_m * k) * (t_m * (t_m * k)))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (l * (l / ((t_m * k) * (t_m * (t_m * k)))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / ((t_m * k) * (t_m * (t_m * k)))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * (l / ((t_m * k) * (t_m * (t_m * k)))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(l / Float64(Float64(t_m * k) * Float64(t_m * Float64(t_m * k))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (l * (l / ((t_m * k) * (t_m * (t_m * k)))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(l * N[(l / N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. 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 52.5

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Applied rewrites 52.5%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   5. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   8. lower-/.f64 56.6

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

   11. *-commutative N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

   12. associate-*l* N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   14. lower-*.f64 58.6

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

7. Applied rewrites 58.6%

   \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

   2. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \cdot \ell \]

   3. associate-*r* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k}} \cdot \ell \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot k} \cdot \ell \]

   5. associate-*l* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right)} \cdot k} \cdot \ell \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\color{blue}{\left(\left(t \cdot t\right) \cdot k\right)} \cdot t\right) \cdot k} \cdot \ell \]

   7. *-commutative N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot k} \cdot \ell \]

   8. *-commutative N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right)} \cdot k} \cdot \ell \]

   9. associate-*l* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(\color{blue}{\left(t \cdot t\right)} \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   13. associate-*l* N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(t \cdot k\right)} \cdot \ell \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   16. lower-*.f64 66.9

       \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

9. Applied rewrites 66.9%

   \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

10. Final simplification 66.9%

    \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot \left(t \cdot k\right)\right)} \]
11. Add Preprocessing

Alternative 22: 58.6% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{\left(t\_m \cdot t\_m\right) \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\right) \end{array} \]

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) (* 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) {
	return t_s * (l * (l / ((t_m * t_m) * (t_m * (k * k)))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (l * (l / ((t_m * t_m) * (t_m * (k * k)))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / ((t_m * t_m) * (t_m * (k * k)))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * (l / ((t_m * t_m) * (t_m * (k * k)))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(l / Float64(Float64(t_m * t_m) * Float64(t_m * Float64(k * k))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (l * (l / ((t_m * t_m) * (t_m * (k * k)))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(l * N[(l / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. 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 52.5

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Applied rewrites 52.5%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   5. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   8. lower-/.f64 56.6

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

   11. *-commutative N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

   12. associate-*l* N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   14. lower-*.f64 58.6

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

7. Applied rewrites 58.6%

   \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]

8. Final simplification 58.6%

   \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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))))