Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.6% → 82.0%

Time: 8.7s

Alternatives: 22

Speedup: 12.5×

• 27.4% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    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: 55.6% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    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: 82.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.62 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + e^{\log \left(\frac{k\_m}{t\_m}\right) \cdot 2}\right) + 1\right)}\\ \mathbf{elif}\;l\_m \leq 3.55 \cdot 10^{+146}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k\_m \cdot t\_m\right)}^{2}, {\left(\sin k\_m \cdot k\_m\right)}^{2}\right)}{\cos k\_m \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log l\_m\right)} \cdot \sin k\_m\right) \cdot \frac{\sin k\_m}{\cos k\_m}\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 1.62e-162)
    (/
     2.0
     (*
      (* (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (sin k_m)) (tan k_m))
      (+ (+ 1.0 (exp (* (log (/ k_m t_m)) 2.0))) 1.0)))
    (if (<= l_m 3.55e+146)
      (/
       2.0
       (*
        (/
         (fma 2.0 (pow (* (sin k_m) t_m) 2.0) (pow (* (sin k_m) k_m) 2.0))
         (* (cos k_m) (* l_m l_m)))
        t_m))
      (/
       2.0
       (*
        (*
         (* (exp (fma (log t_m) 3.0 (* -2.0 (log l_m)))) (sin k_m))
         (/ (sin k_m) (cos k_m)))
        2.0))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (l_m <= 1.62e-162) {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k_m)) * tan(k_m)) * ((1.0 + exp((log((k_m / t_m)) * 2.0))) + 1.0));
	} else if (l_m <= 3.55e+146) {
		tmp = 2.0 / ((fma(2.0, pow((sin(k_m) * t_m), 2.0), pow((sin(k_m) * k_m), 2.0)) / (cos(k_m) * (l_m * l_m))) * t_m);
	} else {
		tmp = 2.0 / (((exp(fma(log(t_m), 3.0, (-2.0 * log(l_m)))) * sin(k_m)) * (sin(k_m) / cos(k_m))) * 2.0);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (l_m <= 1.62e-162)
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + exp(Float64(log(Float64(k_m / t_m)) * 2.0))) + 1.0)));
	elseif (l_m <= 3.55e+146)
		tmp = Float64(2.0 / Float64(Float64(fma(2.0, (Float64(sin(k_m) * t_m) ^ 2.0), (Float64(sin(k_m) * k_m) ^ 2.0)) / Float64(cos(k_m) * Float64(l_m * l_m))) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(log(t_m), 3.0, Float64(-2.0 * log(l_m)))) * sin(k_m)) * Float64(sin(k_m) / cos(k_m))) * 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 1.62e-162], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Exp[N[(N[Log[N[(k$95$m / t$95$m), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l$95$m, 3.55e+146], N[(2.0 / N[(N[(N[(2.0 * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if l < 1.6199999999999999e-162

   1. Initial program 58.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 6.5

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

   4. Applied rewrites 6.5%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lift-/.f64 4.0

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

   6. Applied rewrites 4.0%

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

   if 1.6199999999999999e-162 < l < 3.55e146

   1. Initial program 63.3%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 88.8%

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

   if 3.55e146 < l

   1. Initial program 25.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 27.0

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

   4. Applied rewrites 27.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 27.1

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

   6. Applied rewrites 27.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   7. Step-by-step derivation

      1. lift-tan.f64 N/A

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

      2. tan-quot N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lower-cos.f64 27.1

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

   8. Applied rewrites 27.1%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 30.1%

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

4. Final simplification 29.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.62 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + e^{\log \left(\frac{k}{t}\right) \cdot 2}\right) + 1\right)}\\ \mathbf{elif}\;\ell \leq 3.55 \cdot 10^{+146}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \sin k\right) \cdot \frac{\sin k}{\cos k}\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 0:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\_m\right) \cdot k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;l\_m \cdot l\_m \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k\_m \cdot t\_m\right)}^{2}, {\left(\sin k\_m \cdot k\_m\right)}^{2}\right)}{\cos k\_m \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log l\_m\right)} \cdot \sin k\_m\right) \cdot \frac{\sin k\_m}{\cos k\_m}\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 0.0)
    (/
     2.0
     (*
      (* (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (sin k_m)) k_m)
      (+ (+ 1.0 (pow (/ k_m t_m) 2.0)) 1.0)))
    (if (<= (* l_m l_m) 2e+289)
      (/
       2.0
       (*
        (/
         (fma 2.0 (pow (* (sin k_m) t_m) 2.0) (pow (* (sin k_m) k_m) 2.0))
         (* (cos k_m) (* l_m l_m)))
        t_m))
      (/
       2.0
       (*
        (*
         (* (exp (fma (log t_m) 3.0 (* -2.0 (log l_m)))) (sin k_m))
         (/ (sin k_m) (cos k_m)))
        2.0))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if ((l_m * l_m) <= 0.0) {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k_m)) * k_m) * ((1.0 + pow((k_m / t_m), 2.0)) + 1.0));
	} else if ((l_m * l_m) <= 2e+289) {
		tmp = 2.0 / ((fma(2.0, pow((sin(k_m) * t_m), 2.0), pow((sin(k_m) * k_m), 2.0)) / (cos(k_m) * (l_m * l_m))) * t_m);
	} else {
		tmp = 2.0 / (((exp(fma(log(t_m), 3.0, (-2.0 * log(l_m)))) * sin(k_m)) * (sin(k_m) / cos(k_m))) * 2.0);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 0.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * sin(k_m)) * k_m) * Float64(Float64(1.0 + (Float64(k_m / t_m) ^ 2.0)) + 1.0)));
	elseif (Float64(l_m * l_m) <= 2e+289)
		tmp = Float64(2.0 / Float64(Float64(fma(2.0, (Float64(sin(k_m) * t_m) ^ 2.0), (Float64(sin(k_m) * k_m) ^ 2.0)) / Float64(cos(k_m) * Float64(l_m * l_m))) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(log(t_m), 3.0, Float64(-2.0 * log(l_m)))) * sin(k_m)) * Float64(sin(k_m) / cos(k_m))) * 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 0.0], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 2e+289], N[(2.0 / N[(N[(N[(2.0 * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 54.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 16.0

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

   4. Applied rewrites 16.0%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 16.0%

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

   if 0.0 < (*.f64 l l) < 2.0000000000000001e289

   1. Initial program 66.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.4%

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

   if 2.0000000000000001e289 < (*.f64 l l)

   1. Initial program 32.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 14.5

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

   4. Applied rewrites 14.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 14.6

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

   6. Applied rewrites 14.6%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   7. Step-by-step derivation

      1. lift-tan.f64 N/A

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

      2. tan-quot N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lower-cos.f64 14.6

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

   8. Applied rewrites 14.6%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 16.2%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \sin k\right) \cdot \frac{\sin k}{\cos k}\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 0:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\_m\right) \cdot k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;l\_m \cdot l\_m \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k\_m \cdot t\_m\right)}^{2}, {\left(\sin k\_m \cdot k\_m\right)}^{2}\right)}{\cos k\_m \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log l\_m\right)} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 0.0)
    (/
     2.0
     (*
      (* (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (sin k_m)) k_m)
      (+ (+ 1.0 (pow (/ k_m t_m) 2.0)) 1.0)))
    (if (<= (* l_m l_m) 2e+289)
      (/
       2.0
       (*
        (/
         (fma 2.0 (pow (* (sin k_m) t_m) 2.0) (pow (* (sin k_m) k_m) 2.0))
         (* (cos k_m) (* l_m l_m)))
        t_m))
      (/
       2.0
       (*
        (*
         (* (exp (fma (log t_m) 3.0 (* -2.0 (log l_m)))) (sin k_m))
         (tan k_m))
        (+ 1.0 1.0)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if ((l_m * l_m) <= 0.0) {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k_m)) * k_m) * ((1.0 + pow((k_m / t_m), 2.0)) + 1.0));
	} else if ((l_m * l_m) <= 2e+289) {
		tmp = 2.0 / ((fma(2.0, pow((sin(k_m) * t_m), 2.0), pow((sin(k_m) * k_m), 2.0)) / (cos(k_m) * (l_m * l_m))) * t_m);
	} else {
		tmp = 2.0 / (((exp(fma(log(t_m), 3.0, (-2.0 * log(l_m)))) * sin(k_m)) * tan(k_m)) * (1.0 + 1.0));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 0.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * sin(k_m)) * k_m) * Float64(Float64(1.0 + (Float64(k_m / t_m) ^ 2.0)) + 1.0)));
	elseif (Float64(l_m * l_m) <= 2e+289)
		tmp = Float64(2.0 / Float64(Float64(fma(2.0, (Float64(sin(k_m) * t_m) ^ 2.0), (Float64(sin(k_m) * k_m) ^ 2.0)) / Float64(cos(k_m) * Float64(l_m * l_m))) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(log(t_m), 3.0, Float64(-2.0 * log(l_m)))) * sin(k_m)) * tan(k_m)) * Float64(1.0 + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 0.0], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 2e+289], N[(2.0 / N[(N[(N[(2.0 * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 54.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 16.0

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

   4. Applied rewrites 16.0%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 16.0%

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

   if 0.0 < (*.f64 l l) < 2.0000000000000001e289

   1. Initial program 66.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.4%

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

   if 2.0000000000000001e289 < (*.f64 l l)

   1. Initial program 32.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 14.5

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

   4. Applied rewrites 14.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 14.6

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

   6. Applied rewrites 14.6%

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

   7. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{1} + 1\right)} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 16.2

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

   9. Applied rewrites 16.2%

      \[\leadsto \frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{1} + 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 81.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log l\_m\right)} \cdot \sin k\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 0:\\ \;\;\;\;\frac{2}{\left(t\_2 \cdot k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;l\_m \cdot l\_m \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k\_m \cdot t\_m\right)}^{2}, {\left(\sin k\_m \cdot k\_m\right)}^{2}\right)}{\cos k\_m \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_2 \cdot \tan k\_m\right) \cdot \left(1 + 1\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (let* ((t_2 (* (exp (fma (log t_m) 3.0 (* -2.0 (log l_m)))) (sin k_m))))
   (*
    t_s
    (if (<= (* l_m l_m) 0.0)
      (/ 2.0 (* (* t_2 k_m) (+ (+ 1.0 (pow (/ k_m t_m) 2.0)) 1.0)))
      (if (<= (* l_m l_m) 2e+289)
        (/
         2.0
         (*
          (/
           (fma 2.0 (pow (* (sin k_m) t_m) 2.0) (pow (* (sin k_m) k_m) 2.0))
           (* (cos k_m) (* l_m l_m)))
          t_m))
        (/ 2.0 (* (* t_2 (tan k_m)) (+ 1.0 1.0))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double t_2 = exp(fma(log(t_m), 3.0, (-2.0 * log(l_m)))) * sin(k_m);
	double tmp;
	if ((l_m * l_m) <= 0.0) {
		tmp = 2.0 / ((t_2 * k_m) * ((1.0 + pow((k_m / t_m), 2.0)) + 1.0));
	} else if ((l_m * l_m) <= 2e+289) {
		tmp = 2.0 / ((fma(2.0, pow((sin(k_m) * t_m), 2.0), pow((sin(k_m) * k_m), 2.0)) / (cos(k_m) * (l_m * l_m))) * t_m);
	} else {
		tmp = 2.0 / ((t_2 * tan(k_m)) * (1.0 + 1.0));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	t_2 = Float64(exp(fma(log(t_m), 3.0, Float64(-2.0 * log(l_m)))) * sin(k_m))
	tmp = 0.0
	if (Float64(l_m * l_m) <= 0.0)
		tmp = Float64(2.0 / Float64(Float64(t_2 * k_m) * Float64(Float64(1.0 + (Float64(k_m / t_m) ^ 2.0)) + 1.0)));
	elseif (Float64(l_m * l_m) <= 2e+289)
		tmp = Float64(2.0 / Float64(Float64(fma(2.0, (Float64(sin(k_m) * t_m) ^ 2.0), (Float64(sin(k_m) * k_m) ^ 2.0)) / Float64(cos(k_m) * Float64(l_m * l_m))) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(t_2 * tan(k_m)) * Float64(1.0 + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := Block[{t$95$2 = N[(N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 0.0], N[(2.0 / N[(N[(t$95$2 * k$95$m), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 2e+289], N[(2.0 / N[(N[(N[(2.0 * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$2 * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 54.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 16.0

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

   4. Applied rewrites 16.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 16.0

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

   6. Applied rewrites 16.0%

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

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \sin k\right) \cdot \color{blue}{k}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 16.0%

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

   if 0.0 < (*.f64 l l) < 2.0000000000000001e289

   1. Initial program 66.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.4%

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

   if 2.0000000000000001e289 < (*.f64 l l)

   1. Initial program 32.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 14.5

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

   4. Applied rewrites 14.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 14.6

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

   6. Applied rewrites 14.6%

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

   7. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{1} + 1\right)} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 16.2

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

   9. Applied rewrites 16.2%

      \[\leadsto \frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{1} + 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 79.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m \cdot t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}\right) \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;l\_m \leq 3.55 \cdot 10^{+146}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k\_m \cdot t\_m\right)}^{2}, {\left(\sin k\_m \cdot k\_m\right)}^{2}\right)}{\cos k\_m \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log l\_m\right)} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 1.6e-162)
    (/
     2.0
     (*
      (* (* (* (/ (* t_m t_m) l_m) (/ t_m l_m)) (sin k_m)) (tan k_m))
      (+ (+ 1.0 (pow (/ k_m t_m) 2.0)) 1.0)))
    (if (<= l_m 3.55e+146)
      (/
       2.0
       (*
        (/
         (fma 2.0 (pow (* (sin k_m) t_m) 2.0) (pow (* (sin k_m) k_m) 2.0))
         (* (cos k_m) (* l_m l_m)))
        t_m))
      (/
       2.0
       (*
        (*
         (* (exp (fma (log t_m) 3.0 (* -2.0 (log l_m)))) (sin k_m))
         (tan k_m))
        (+ 1.0 1.0)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (l_m <= 1.6e-162) {
		tmp = 2.0 / ((((((t_m * t_m) / l_m) * (t_m / l_m)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t_m), 2.0)) + 1.0));
	} else if (l_m <= 3.55e+146) {
		tmp = 2.0 / ((fma(2.0, pow((sin(k_m) * t_m), 2.0), pow((sin(k_m) * k_m), 2.0)) / (cos(k_m) * (l_m * l_m))) * t_m);
	} else {
		tmp = 2.0 / (((exp(fma(log(t_m), 3.0, (-2.0 * log(l_m)))) * sin(k_m)) * tan(k_m)) * (1.0 + 1.0));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (l_m <= 1.6e-162)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l_m) * Float64(t_m / l_m)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t_m) ^ 2.0)) + 1.0)));
	elseif (l_m <= 3.55e+146)
		tmp = Float64(2.0 / Float64(Float64(fma(2.0, (Float64(sin(k_m) * t_m) ^ 2.0), (Float64(sin(k_m) * k_m) ^ 2.0)) / Float64(cos(k_m) * Float64(l_m * l_m))) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(log(t_m), 3.0, Float64(-2.0 * log(l_m)))) * sin(k_m)) * tan(k_m)) * Float64(1.0 + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 1.6e-162], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l$95$m, 3.55e+146], N[(2.0 / N[(N[(N[(2.0 * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if l < 1.59999999999999988e-162

   1. Initial program 58.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 6.5

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

   4. Applied rewrites 6.5%

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

      1. lift-exp.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. exp-diff N/A

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

      8. pow-to-exp N/A

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

      9. pow-to-exp N/A

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

      10. pow3 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 73.1

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

   6. Applied rewrites 73.1%

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

   if 1.59999999999999988e-162 < l < 3.55e146

   1. Initial program 63.3%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 88.8%

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

   if 3.55e146 < l

   1. Initial program 25.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 27.0

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

   4. Applied rewrites 27.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 27.1

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

   6. Applied rewrites 27.1%

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

   7. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{1} + 1\right)} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 30.1

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

   9. Applied rewrites 30.1%

      \[\leadsto \frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{1} + 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 76.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{-104}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot k\_m}{l\_m \cdot l\_m} \cdot \frac{{\sin k\_m}^{2} \cdot t\_m}{\cos k\_m}}\\ \mathbf{elif}\;t\_m \leq 7.2 \cdot 10^{+169}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m \cdot t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}\right) \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log l\_m\right)} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5e-104)
    (/
     2.0
     (* (/ (* k_m k_m) (* l_m l_m)) (/ (* (pow (sin k_m) 2.0) t_m) (cos k_m))))
    (if (<= t_m 7.2e+169)
      (/
       2.0
       (*
        (* (* (* (/ (* t_m t_m) l_m) (/ t_m l_m)) (sin k_m)) (tan k_m))
        (+ (+ 1.0 (pow (/ k_m t_m) 2.0)) 1.0)))
      (/
       2.0
       (*
        (*
         (* (exp (fma (log t_m) 3.0 (* -2.0 (log l_m)))) (sin k_m))
         (tan k_m))
        (+ 1.0 1.0)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 5e-104) {
		tmp = 2.0 / (((k_m * k_m) / (l_m * l_m)) * ((pow(sin(k_m), 2.0) * t_m) / cos(k_m)));
	} else if (t_m <= 7.2e+169) {
		tmp = 2.0 / ((((((t_m * t_m) / l_m) * (t_m / l_m)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t_m), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (((exp(fma(log(t_m), 3.0, (-2.0 * log(l_m)))) * sin(k_m)) * tan(k_m)) * (1.0 + 1.0));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 5e-104)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) / Float64(l_m * l_m)) * Float64(Float64((sin(k_m) ^ 2.0) * t_m) / cos(k_m))));
	elseif (t_m <= 7.2e+169)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l_m) * Float64(t_m / l_m)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t_m) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(log(t_m), 3.0, Float64(-2.0 * log(l_m)))) * sin(k_m)) * tan(k_m)) * Float64(1.0 + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 5e-104], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.2e+169], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.99999999999999979e-104

   1. Initial program 50.7%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 58.9

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

   5. Applied rewrites 58.9%

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

   if 4.99999999999999979e-104 < t < 7.20000000000000019e169

   1. Initial program 70.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 48.4

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

   4. Applied rewrites 48.4%

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

      1. lift-exp.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. exp-diff N/A

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

      8. pow-to-exp N/A

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

      9. pow-to-exp N/A

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

      10. pow3 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 82.3

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

   6. Applied rewrites 82.3%

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

   if 7.20000000000000019e169 < t

   1. Initial program 58.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 45.0

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

   4. Applied rewrites 45.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 45.1

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

   6. Applied rewrites 45.1%

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

   7. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{1} + 1\right)} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 45.1

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

   9. Applied rewrites 45.1%

      \[\leadsto \frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{1} + 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 76.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{-104}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot k\_m}{l\_m \cdot l\_m} \cdot \frac{{\sin k\_m}^{2} \cdot t\_m}{\cos k\_m}}\\ \mathbf{elif}\;t\_m \leq 7.2 \cdot 10^{+169}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m \cdot t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}\right) \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log l\_m, -2, \log t\_m \cdot 3\right)} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5e-104)
    (/
     2.0
     (* (/ (* k_m k_m) (* l_m l_m)) (/ (* (pow (sin k_m) 2.0) t_m) (cos k_m))))
    (if (<= t_m 7.2e+169)
      (/
       2.0
       (*
        (* (* (* (/ (* t_m t_m) l_m) (/ t_m l_m)) (sin k_m)) (tan k_m))
        (+ (+ 1.0 (pow (/ k_m t_m) 2.0)) 1.0)))
      (/
       2.0
       (*
        (*
         (* (exp (fma (log l_m) -2.0 (* (log t_m) 3.0))) (sin k_m))
         (tan k_m))
        (+ 1.0 1.0)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 5e-104) {
		tmp = 2.0 / (((k_m * k_m) / (l_m * l_m)) * ((pow(sin(k_m), 2.0) * t_m) / cos(k_m)));
	} else if (t_m <= 7.2e+169) {
		tmp = 2.0 / ((((((t_m * t_m) / l_m) * (t_m / l_m)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t_m), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (((exp(fma(log(l_m), -2.0, (log(t_m) * 3.0))) * sin(k_m)) * tan(k_m)) * (1.0 + 1.0));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 5e-104)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) / Float64(l_m * l_m)) * Float64(Float64((sin(k_m) ^ 2.0) * t_m) / cos(k_m))));
	elseif (t_m <= 7.2e+169)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l_m) * Float64(t_m / l_m)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t_m) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(log(l_m), -2.0, Float64(log(t_m) * 3.0))) * sin(k_m)) * tan(k_m)) * Float64(1.0 + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 5e-104], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.2e+169], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[Log[l$95$m], $MachinePrecision] * -2.0 + N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.99999999999999979e-104

   1. Initial program 50.7%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 58.9

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

   5. Applied rewrites 58.9%

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

   if 4.99999999999999979e-104 < t < 7.20000000000000019e169

   1. Initial program 70.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 48.4

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

   4. Applied rewrites 48.4%

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

      1. lift-exp.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. exp-diff N/A

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

      8. pow-to-exp N/A

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

      9. pow-to-exp N/A

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

      10. pow3 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 82.3

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

   6. Applied rewrites 82.3%

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

   if 7.20000000000000019e169 < t

   1. Initial program 58.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 45.0

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

   4. Applied rewrites 45.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 45.1

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

   6. Applied rewrites 45.1%

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

   7. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{1} + 1\right)} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 45.1

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

   9. Applied rewrites 45.1%

      \[\leadsto \frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{1} + 1\right)} \]
   10. Step-by-step derivation

       1. lift-log.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 N/A

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

       5. lift-log.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. lift-log.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lift-log.f64 45.0

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

   11. Applied rewrites 45.0%

       \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\mathsf{fma}\left(\log \ell, -2, \log t \cdot 3\right)}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{-104}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot k\_m}{l\_m \cdot l\_m} \cdot \frac{{\sin k\_m}^{2} \cdot t\_m}{\cos k\_m}}\\ \mathbf{elif}\;t\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m \cdot t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}\right) \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log l\_m\right)} \cdot \left(\mathsf{fma}\left(-0.16666666666666666, k\_m \cdot k\_m, 1\right) \cdot k\_m\right)\right) \cdot \tan k\_m\right) \cdot \left(1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5e-104)
    (/
     2.0
     (* (/ (* k_m k_m) (* l_m l_m)) (/ (* (pow (sin k_m) 2.0) t_m) (cos k_m))))
    (if (<= t_m 1.35e+154)
      (/
       2.0
       (*
        (* (* (* (/ (* t_m t_m) l_m) (/ t_m l_m)) (sin k_m)) (tan k_m))
        (+ (+ 1.0 (pow (/ k_m t_m) 2.0)) 1.0)))
      (/
       2.0
       (*
        (*
         (*
          (exp (fma (log t_m) 3.0 (* -2.0 (log l_m))))
          (* (fma -0.16666666666666666 (* k_m k_m) 1.0) k_m))
         (tan k_m))
        (+ 1.0 1.0)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 5e-104) {
		tmp = 2.0 / (((k_m * k_m) / (l_m * l_m)) * ((pow(sin(k_m), 2.0) * t_m) / cos(k_m)));
	} else if (t_m <= 1.35e+154) {
		tmp = 2.0 / ((((((t_m * t_m) / l_m) * (t_m / l_m)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t_m), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (((exp(fma(log(t_m), 3.0, (-2.0 * log(l_m)))) * (fma(-0.16666666666666666, (k_m * k_m), 1.0) * k_m)) * tan(k_m)) * (1.0 + 1.0));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 5e-104)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) / Float64(l_m * l_m)) * Float64(Float64((sin(k_m) ^ 2.0) * t_m) / cos(k_m))));
	elseif (t_m <= 1.35e+154)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l_m) * Float64(t_m / l_m)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t_m) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(log(t_m), 3.0, Float64(-2.0 * log(l_m)))) * Float64(fma(-0.16666666666666666, Float64(k_m * k_m), 1.0) * k_m)) * tan(k_m)) * Float64(1.0 + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 5e-104], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.35e+154], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.99999999999999979e-104

   1. Initial program 50.7%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 58.9

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

   5. Applied rewrites 58.9%

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

   if 4.99999999999999979e-104 < t < 1.35000000000000003e154

   1. Initial program 74.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 46.3

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

   4. Applied rewrites 46.3%

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

      1. lift-exp.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. exp-diff N/A

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

      8. pow-to-exp N/A

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

      9. pow-to-exp N/A

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

      10. pow3 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 84.9

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

   6. Applied rewrites 84.9%

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

   if 1.35000000000000003e154 < t

   1. Initial program 52.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 50.2

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

   4. Applied rewrites 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 50.3

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

   6. Applied rewrites 50.3%

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

   7. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{1} + 1\right)} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 45.7

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

   9. Applied rewrites 45.7%

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

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \color{blue}{\left(k \cdot \left(1 + \frac{-1}{6} \cdot {k}^{2}\right)\right)}\right) \cdot \tan k\right) \cdot \left(1 + 1\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \left(\left(1 + \frac{-1}{6} \cdot {k}^{2}\right) \cdot \color{blue}{k}\right)\right) \cdot \tan k\right) \cdot \left(1 + 1\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \left(\left(1 + \frac{-1}{6} \cdot {k}^{2}\right) \cdot \color{blue}{k}\right)\right) \cdot \tan k\right) \cdot \left(1 + 1\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \left(\left(\frac{-1}{6} \cdot {k}^{2} + 1\right) \cdot k\right)\right) \cdot \tan k\right) \cdot \left(1 + 1\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \left(\mathsf{fma}\left(\frac{-1}{6}, {k}^{2}, 1\right) \cdot k\right)\right) \cdot \tan k\right) \cdot \left(1 + 1\right)} \]

       5. pow2 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \left(\mathsf{fma}\left(\frac{-1}{6}, k \cdot k, 1\right) \cdot k\right)\right) \cdot \tan k\right) \cdot \left(1 + 1\right)} \]

       6. lift-*.f64 42.4

          \[\leadsto \frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \left(\mathsf{fma}\left(-0.16666666666666666, k \cdot k, 1\right) \cdot k\right)\right) \cdot \tan k\right) \cdot \left(1 + 1\right)} \]

   12. Applied rewrites 42.4%

       \[\leadsto \frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(-0.16666666666666666, k \cdot k, 1\right) \cdot k\right)}\right) \cdot \tan k\right) \cdot \left(1 + 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{-104}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot k\_m}{l\_m \cdot l\_m} \cdot \frac{{\sin k\_m}^{2} \cdot t\_m}{\cos k\_m}}\\ \mathbf{elif}\;t\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m \cdot t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}\right) \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log l\_m\right)} \cdot k\_m\right) \cdot \tan k\_m\right) \cdot \left(1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5e-104)
    (/
     2.0
     (* (/ (* k_m k_m) (* l_m l_m)) (/ (* (pow (sin k_m) 2.0) t_m) (cos k_m))))
    (if (<= t_m 1.35e+154)
      (/
       2.0
       (*
        (* (* (* (/ (* t_m t_m) l_m) (/ t_m l_m)) (sin k_m)) (tan k_m))
        (+ (+ 1.0 (pow (/ k_m t_m) 2.0)) 1.0)))
      (/
       2.0
       (*
        (* (* (exp (fma (log t_m) 3.0 (* -2.0 (log l_m)))) k_m) (tan k_m))
        (+ 1.0 1.0)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 5e-104) {
		tmp = 2.0 / (((k_m * k_m) / (l_m * l_m)) * ((pow(sin(k_m), 2.0) * t_m) / cos(k_m)));
	} else if (t_m <= 1.35e+154) {
		tmp = 2.0 / ((((((t_m * t_m) / l_m) * (t_m / l_m)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t_m), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (((exp(fma(log(t_m), 3.0, (-2.0 * log(l_m)))) * k_m) * tan(k_m)) * (1.0 + 1.0));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 5e-104)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) / Float64(l_m * l_m)) * Float64(Float64((sin(k_m) ^ 2.0) * t_m) / cos(k_m))));
	elseif (t_m <= 1.35e+154)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l_m) * Float64(t_m / l_m)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t_m) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(log(t_m), 3.0, Float64(-2.0 * log(l_m)))) * k_m) * tan(k_m)) * Float64(1.0 + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 5e-104], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.35e+154], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * k$95$m), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.99999999999999979e-104

   1. Initial program 50.7%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 58.9

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

   5. Applied rewrites 58.9%

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

   if 4.99999999999999979e-104 < t < 1.35000000000000003e154

   1. Initial program 74.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 46.3

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

   4. Applied rewrites 46.3%

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

      1. lift-exp.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. exp-diff N/A

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

      8. pow-to-exp N/A

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

      9. pow-to-exp N/A

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

      10. pow3 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 84.9

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

   6. Applied rewrites 84.9%

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

   if 1.35000000000000003e154 < t

   1. Initial program 52.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 50.2

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

   4. Applied rewrites 50.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 50.3

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

   6. Applied rewrites 50.3%

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

   7. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{1} + 1\right)} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 45.7

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

   9. Applied rewrites 45.7%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 42.0%

       \[\leadsto \frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \color{blue}{k}\right) \cdot \tan k\right) \cdot \left(1 + 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 74.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{-104}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot k\_m}{l\_m \cdot l\_m} \cdot \frac{{\sin k\_m}^{2} \cdot t\_m}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m \cdot t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}\right) \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5e-104)
    (/
     2.0
     (* (/ (* k_m k_m) (* l_m l_m)) (/ (* (pow (sin k_m) 2.0) t_m) (cos k_m))))
    (/
     2.0
     (*
      (* (* (* (/ (* t_m t_m) l_m) (/ t_m l_m)) (sin k_m)) (tan k_m))
      (+ (+ 1.0 (pow (/ k_m t_m) 2.0)) 1.0))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 5e-104) {
		tmp = 2.0 / (((k_m * k_m) / (l_m * l_m)) * ((pow(sin(k_m), 2.0) * t_m) / cos(k_m)));
	} else {
		tmp = 2.0 / ((((((t_m * t_m) / l_m) * (t_m / l_m)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t_m), 2.0)) + 1.0));
	}
	return t_s * tmp;
}

Fortran

l_m =     private
k_m =     private
t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 5d-104) then
        tmp = 2.0d0 / (((k_m * k_m) / (l_m * l_m)) * (((sin(k_m) ** 2.0d0) * t_m) / cos(k_m)))
    else
        tmp = 2.0d0 / ((((((t_m * t_m) / l_m) * (t_m / l_m)) * sin(k_m)) * tan(k_m)) * ((1.0d0 + ((k_m / t_m) ** 2.0d0)) + 1.0d0))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 5e-104) {
		tmp = 2.0 / (((k_m * k_m) / (l_m * l_m)) * ((Math.pow(Math.sin(k_m), 2.0) * t_m) / Math.cos(k_m)));
	} else {
		tmp = 2.0 / ((((((t_m * t_m) / l_m) * (t_m / l_m)) * Math.sin(k_m)) * Math.tan(k_m)) * ((1.0 + Math.pow((k_m / t_m), 2.0)) + 1.0));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if t_m <= 5e-104:
		tmp = 2.0 / (((k_m * k_m) / (l_m * l_m)) * ((math.pow(math.sin(k_m), 2.0) * t_m) / math.cos(k_m)))
	else:
		tmp = 2.0 / ((((((t_m * t_m) / l_m) * (t_m / l_m)) * math.sin(k_m)) * math.tan(k_m)) * ((1.0 + math.pow((k_m / t_m), 2.0)) + 1.0))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 5e-104)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) / Float64(l_m * l_m)) * Float64(Float64((sin(k_m) ^ 2.0) * t_m) / cos(k_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l_m) * Float64(t_m / l_m)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t_m) ^ 2.0)) + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (t_m <= 5e-104)
		tmp = 2.0 / (((k_m * k_m) / (l_m * l_m)) * (((sin(k_m) ^ 2.0) * t_m) / cos(k_m)));
	else
		tmp = 2.0 / ((((((t_m * t_m) / l_m) * (t_m / l_m)) * sin(k_m)) * tan(k_m)) * ((1.0 + ((k_m / t_m) ^ 2.0)) + 1.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 5e-104], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.99999999999999979e-104

   1. Initial program 50.7%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 58.9

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

   5. Applied rewrites 58.9%

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

   if 4.99999999999999979e-104 < t

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 47.5

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

   4. Applied rewrites 47.5%

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

      1. lift-exp.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. exp-diff N/A

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

      8. pow-to-exp N/A

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

      9. pow-to-exp N/A

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

      10. pow3 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 76.5

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

   6. Applied rewrites 76.5%

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

Alternative 11: 72.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.25 \cdot 10^{-102}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot k\_m}{l\_m \cdot l\_m} \cdot \frac{{\sin k\_m}^{2} \cdot t\_m}{\cos k\_m}}\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{l\_m \cdot l\_m} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \mathsf{fma}\left(\frac{k\_m}{t\_m}, \frac{k\_m}{t\_m}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{e^{\mathsf{fma}\left(\log t\_m, 3, \log k\_m \cdot 2\right)}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.25e-102)
    (/
     2.0
     (* (/ (* k_m k_m) (* l_m l_m)) (/ (* (pow (sin k_m) 2.0) t_m) (cos k_m))))
    (if (<= t_m 1.05e+102)
      (/
       2.0
       (*
        (* (* (/ (* (* t_m t_m) t_m) (* l_m l_m)) (sin k_m)) (tan k_m))
        (fma (/ k_m t_m) (/ k_m t_m) 2.0)))
      (/ (* l_m l_m) (exp (fma (log t_m) 3.0 (* (log k_m) 2.0))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 2.25e-102) {
		tmp = 2.0 / (((k_m * k_m) / (l_m * l_m)) * ((pow(sin(k_m), 2.0) * t_m) / cos(k_m)));
	} else if (t_m <= 1.05e+102) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * sin(k_m)) * tan(k_m)) * fma((k_m / t_m), (k_m / t_m), 2.0));
	} else {
		tmp = (l_m * l_m) / exp(fma(log(t_m), 3.0, (log(k_m) * 2.0)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 2.25e-102)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) / Float64(l_m * l_m)) * Float64(Float64((sin(k_m) ^ 2.0) * t_m) / cos(k_m))));
	elseif (t_m <= 1.05e+102)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / Float64(l_m * l_m)) * sin(k_m)) * tan(k_m)) * fma(Float64(k_m / t_m), Float64(k_m / t_m), 2.0)));
	else
		tmp = Float64(Float64(l_m * l_m) / exp(fma(log(t_m), 3.0, Float64(log(k_m) * 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.25e-102], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.05e+102], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.25e-102

   1. Initial program 50.7%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 58.9

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

   5. Applied rewrites 58.9%

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

   if 2.25e-102 < t < 1.05000000000000001e102

   1. Initial program 82.6%

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

      1. 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. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 82.6

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

   4. Applied rewrites 82.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 82.6

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

   6. Applied rewrites 82.6%

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

   if 1.05000000000000001e102 < t

   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

   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

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 40.0

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

   5. Applied rewrites 40.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. pow-to-exp N/A

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

      7. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{e^{3 \cdot \log t} \cdot {k}^{2}} \]

      8. pow-to-exp N/A

         \[\leadsto \frac{\ell \cdot \ell}{e^{3 \cdot \log t} \cdot e^{\log k \cdot 2}} \]

      9. prod-exp N/A

         \[\leadsto \frac{\ell \cdot \ell}{e^{3 \cdot \log t + \log k \cdot 2}} \]

      10. lower-exp.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{e^{3 \cdot \log t + \log k \cdot 2}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell \cdot \ell}{e^{\log t \cdot 3 + \log k \cdot 2}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]

      13. lift-log.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]

      15. lower-log.f64 29.5

          \[\leadsto \frac{\ell \cdot \ell}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]

   7. Applied rewrites 29.5%

      \[\leadsto \frac{\ell \cdot \ell}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 72.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.25 \cdot 10^{-102}:\\ \;\;\;\;\left(\frac{l\_m \cdot l\_m}{k\_m \cdot k\_m} \cdot \frac{\cos k\_m}{{\sin k\_m}^{2} \cdot t\_m}\right) \cdot 2\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{l\_m \cdot l\_m} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \mathsf{fma}\left(\frac{k\_m}{t\_m}, \frac{k\_m}{t\_m}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{e^{\mathsf{fma}\left(\log t\_m, 3, \log k\_m \cdot 2\right)}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.25e-102)
    (*
     (* (/ (* l_m l_m) (* k_m k_m)) (/ (cos k_m) (* (pow (sin k_m) 2.0) t_m)))
     2.0)
    (if (<= t_m 1.05e+102)
      (/
       2.0
       (*
        (* (* (/ (* (* t_m t_m) t_m) (* l_m l_m)) (sin k_m)) (tan k_m))
        (fma (/ k_m t_m) (/ k_m t_m) 2.0)))
      (/ (* l_m l_m) (exp (fma (log t_m) 3.0 (* (log k_m) 2.0))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 2.25e-102) {
		tmp = (((l_m * l_m) / (k_m * k_m)) * (cos(k_m) / (pow(sin(k_m), 2.0) * t_m))) * 2.0;
	} else if (t_m <= 1.05e+102) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * sin(k_m)) * tan(k_m)) * fma((k_m / t_m), (k_m / t_m), 2.0));
	} else {
		tmp = (l_m * l_m) / exp(fma(log(t_m), 3.0, (log(k_m) * 2.0)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 2.25e-102)
		tmp = Float64(Float64(Float64(Float64(l_m * l_m) / Float64(k_m * k_m)) * Float64(cos(k_m) / Float64((sin(k_m) ^ 2.0) * t_m))) * 2.0);
	elseif (t_m <= 1.05e+102)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / Float64(l_m * l_m)) * sin(k_m)) * tan(k_m)) * fma(Float64(k_m / t_m), Float64(k_m / t_m), 2.0)));
	else
		tmp = Float64(Float64(l_m * l_m) / exp(fma(log(t_m), 3.0, Float64(log(k_m) * 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.25e-102], N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$m, 1.05e+102], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.25e-102

   1. Initial program 50.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 58.6%

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

   if 2.25e-102 < t < 1.05000000000000001e102

   1. Initial program 82.6%

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

      1. 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. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 82.6

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

   4. Applied rewrites 82.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{k}{t}}, \frac{k}{t}, 2\right)} \]

      11. lift-/.f64 82.6

          \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \color{blue}{\frac{k}{t}}, 2\right)} \]

   6. Applied rewrites 82.6%

      \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   if 1.05000000000000001e102 < t

   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

   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

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 40.0

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 40.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      4. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{\ell \cdot \ell}{e^{\log t \cdot 3} \cdot {\color{blue}{k}}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{e^{3 \cdot \log t} \cdot {k}^{2}} \]

      8. pow-to-exp N/A

         \[\leadsto \frac{\ell \cdot \ell}{e^{3 \cdot \log t} \cdot e^{\log k \cdot 2}} \]

      9. prod-exp N/A

         \[\leadsto \frac{\ell \cdot \ell}{e^{3 \cdot \log t + \log k \cdot 2}} \]

      10. lower-exp.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{e^{3 \cdot \log t + \log k \cdot 2}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell \cdot \ell}{e^{\log t \cdot 3 + \log k \cdot 2}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]

      13. lift-log.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]

      15. lower-log.f64 29.5

          \[\leadsto \frac{\ell \cdot \ell}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]

   7. Applied rewrites 29.5%

      \[\leadsto \frac{\ell \cdot \ell}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 68.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.42 \cdot 10^{-186}:\\ \;\;\;\;\frac{1}{\left(\tan k\_m \cdot \sin k\_m\right) \cdot \frac{\frac{{t\_m}^{3}}{l\_m}}{l\_m}}\\ \mathbf{elif}\;t\_m \leq 3.45 \cdot 10^{-137}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, {t\_m}^{3}, t\_m\right), k\_m \cdot k\_m, 2 \cdot {t\_m}^{3}\right)}{l\_m \cdot l\_m} \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{elif}\;t\_m \leq 2.25 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k\_m}{t\_m} \cdot \frac{k\_m}{t\_m} + 1\right) + 1\right) \cdot \left(\left(\frac{t\_m \cdot t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k\_m \cdot k\_m, 0.16666666666666666\right), k\_m \cdot k\_m, 1\right) \cdot \left(k\_m \cdot k\_m\right)\right)\right)}\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{l\_m \cdot l\_m} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \mathsf{fma}\left(\frac{k\_m}{t\_m}, \frac{k\_m}{t\_m}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{e^{\mathsf{fma}\left(\log t\_m, 3, \log k\_m \cdot 2\right)}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.42e-186)
    (/ 1.0 (* (* (tan k_m) (sin k_m)) (/ (/ (pow t_m 3.0) l_m) l_m)))
    (if (<= t_m 3.45e-137)
      (/
       2.0
       (*
        (/
         (fma
          (fma 0.3333333333333333 (pow t_m 3.0) t_m)
          (* k_m k_m)
          (* 2.0 (pow t_m 3.0)))
         (* l_m l_m))
        (* k_m k_m)))
      (if (<= t_m 2.25e-69)
        (/
         2.0
         (*
          (+ (+ (* (/ k_m t_m) (/ k_m t_m)) 1.0) 1.0)
          (*
           (* (/ (* t_m t_m) l_m) (/ t_m l_m))
           (*
            (fma
             (fma 0.08611111111111111 (* k_m k_m) 0.16666666666666666)
             (* k_m k_m)
             1.0)
            (* k_m k_m)))))
        (if (<= t_m 1.05e+102)
          (/
           2.0
           (*
            (* (* (/ (* (* t_m t_m) t_m) (* l_m l_m)) (sin k_m)) (tan k_m))
            (fma (/ k_m t_m) (/ k_m t_m) 2.0)))
          (/ (* l_m l_m) (exp (fma (log t_m) 3.0 (* (log k_m) 2.0))))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 1.42e-186) {
		tmp = 1.0 / ((tan(k_m) * sin(k_m)) * ((pow(t_m, 3.0) / l_m) / l_m));
	} else if (t_m <= 3.45e-137) {
		tmp = 2.0 / ((fma(fma(0.3333333333333333, pow(t_m, 3.0), t_m), (k_m * k_m), (2.0 * pow(t_m, 3.0))) / (l_m * l_m)) * (k_m * k_m));
	} else if (t_m <= 2.25e-69) {
		tmp = 2.0 / (((((k_m / t_m) * (k_m / t_m)) + 1.0) + 1.0) * ((((t_m * t_m) / l_m) * (t_m / l_m)) * (fma(fma(0.08611111111111111, (k_m * k_m), 0.16666666666666666), (k_m * k_m), 1.0) * (k_m * k_m))));
	} else if (t_m <= 1.05e+102) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * sin(k_m)) * tan(k_m)) * fma((k_m / t_m), (k_m / t_m), 2.0));
	} else {
		tmp = (l_m * l_m) / exp(fma(log(t_m), 3.0, (log(k_m) * 2.0)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 1.42e-186)
		tmp = Float64(1.0 / Float64(Float64(tan(k_m) * sin(k_m)) * Float64(Float64((t_m ^ 3.0) / l_m) / l_m)));
	elseif (t_m <= 3.45e-137)
		tmp = Float64(2.0 / Float64(Float64(fma(fma(0.3333333333333333, (t_m ^ 3.0), t_m), Float64(k_m * k_m), Float64(2.0 * (t_m ^ 3.0))) / Float64(l_m * l_m)) * Float64(k_m * k_m)));
	elseif (t_m <= 2.25e-69)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k_m / t_m) * Float64(k_m / t_m)) + 1.0) + 1.0) * Float64(Float64(Float64(Float64(t_m * t_m) / l_m) * Float64(t_m / l_m)) * Float64(fma(fma(0.08611111111111111, Float64(k_m * k_m), 0.16666666666666666), Float64(k_m * k_m), 1.0) * Float64(k_m * k_m)))));
	elseif (t_m <= 1.05e+102)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / Float64(l_m * l_m)) * sin(k_m)) * tan(k_m)) * fma(Float64(k_m / t_m), Float64(k_m / t_m), 2.0)));
	else
		tmp = Float64(Float64(l_m * l_m) / exp(fma(log(t_m), 3.0, Float64(log(k_m) * 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.42e-186], N[(1.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.45e-137], N[(2.0 / N[(N[(N[(N[(0.3333333333333333 * N[Power[t$95$m, 3.0], $MachinePrecision] + t$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(2.0 * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.25e-69], N[(2.0 / N[(N[(N[(N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.08611111111111111 * N[(k$95$m * k$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.05e+102], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if t < 1.4199999999999999e-186

   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
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t} \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-log.f64 1.2

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 1.2%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\left(\tan k \cdot \sin k\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

   7. Taylor expanded in t around inf

      \[\leadsto \frac{\color{blue}{1}}{\left(\tan k \cdot \sin k\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}} \]
   8. Step-by-step derivation

   9. Applied rewrites 55.4%

      \[\leadsto \frac{\color{blue}{1}}{\left(\tan k \cdot \sin k\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}} \]

   if 1.4199999999999999e-186 < t < 3.44999999999999988e-137

   1. Initial program 45.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

   5. Applied rewrites 56.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, {t}^{3}, t\right), k \cdot k, 2 \cdot {t}^{3}\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   if 3.44999999999999988e-137 < t < 2.25000000000000005e-69

   1. Initial program 42.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left({k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) + 1\right) \cdot {\color{blue}{k}}^{2}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}, {k}^{2}, 1\right) \cdot {\color{blue}{k}}^{2}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{31}{360} \cdot {k}^{2} + \frac{1}{6}, {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, {k}^{2}, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      10. pow2 N/A

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)\right)} \]

      12. pow2 N/A

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)\right)} \]

      13. lift-*.f64 50.4

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)\right)} \]

   7. Applied rewrites 50.4%

      \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left(\frac{k}{t}\right)}}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t}} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      6. lift-/.f64 50.4

         \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \color{blue}{\frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

   9. Applied rewrites 50.4%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       3. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       4. associate-/r* N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       5. pow3 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       6. pow2 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\color{blue}{{t}^{2}} \cdot t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       7. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       9. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\color{blue}{\frac{{t}^{2}}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       10. pow2 N/A

           \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       11. lift-*.f64 N/A

           \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       12. lower-/.f64 58.7

           \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

   11. Applied rewrites 58.7%

       \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

   if 2.25000000000000005e-69 < t < 1.05000000000000001e102

   1. Initial program 86.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-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. unpow3 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{2}} \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{2} \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right)} \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-*.f64 86.8

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right)} \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 86.8%

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left({\color{blue}{\left(\frac{k}{t}\right)}}^{2} + 2\right)} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 2\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 2\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{k}{t}}, \frac{k}{t}, 2\right)} \]

      11. lift-/.f64 86.8

          \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \color{blue}{\frac{k}{t}}, 2\right)} \]

   6. Applied rewrites 86.8%

      \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   if 1.05000000000000001e102 < t

   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

   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

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 40.0

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 40.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      4. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{\ell \cdot \ell}{e^{\log t \cdot 3} \cdot {\color{blue}{k}}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{e^{3 \cdot \log t} \cdot {k}^{2}} \]

      8. pow-to-exp N/A

         \[\leadsto \frac{\ell \cdot \ell}{e^{3 \cdot \log t} \cdot e^{\log k \cdot 2}} \]

      9. prod-exp N/A

         \[\leadsto \frac{\ell \cdot \ell}{e^{3 \cdot \log t + \log k \cdot 2}} \]

      10. lower-exp.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{e^{3 \cdot \log t + \log k \cdot 2}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell \cdot \ell}{e^{\log t \cdot 3 + \log k \cdot 2}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]

      13. lift-log.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]

      15. lower-log.f64 29.5

          \[\leadsto \frac{\ell \cdot \ell}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]

   7. Applied rewrites 29.5%

      \[\leadsto \frac{\ell \cdot \ell}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 14: 69.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.45 \cdot 10^{-137}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, {t\_m}^{3}, t\_m\right), k\_m \cdot k\_m, 2 \cdot {t\_m}^{3}\right)}{l\_m \cdot l\_m} \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{elif}\;t\_m \leq 2.25 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k\_m}{t\_m} \cdot \frac{k\_m}{t\_m} + 1\right) + 1\right) \cdot \left(\left(\frac{t\_m \cdot t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k\_m \cdot k\_m, 0.16666666666666666\right), k\_m \cdot k\_m, 1\right) \cdot \left(k\_m \cdot k\_m\right)\right)\right)}\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{l\_m \cdot l\_m} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \mathsf{fma}\left(\frac{k\_m}{t\_m}, \frac{k\_m}{t\_m}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{e^{\mathsf{fma}\left(\log t\_m, 3, \log k\_m \cdot 2\right)}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.45e-137)
    (/
     2.0
     (*
      (/
       (fma
        (fma 0.3333333333333333 (pow t_m 3.0) t_m)
        (* k_m k_m)
        (* 2.0 (pow t_m 3.0)))
       (* l_m l_m))
      (* k_m k_m)))
    (if (<= t_m 2.25e-69)
      (/
       2.0
       (*
        (+ (+ (* (/ k_m t_m) (/ k_m t_m)) 1.0) 1.0)
        (*
         (* (/ (* t_m t_m) l_m) (/ t_m l_m))
         (*
          (fma
           (fma 0.08611111111111111 (* k_m k_m) 0.16666666666666666)
           (* k_m k_m)
           1.0)
          (* k_m k_m)))))
      (if (<= t_m 1.05e+102)
        (/
         2.0
         (*
          (* (* (/ (* (* t_m t_m) t_m) (* l_m l_m)) (sin k_m)) (tan k_m))
          (fma (/ k_m t_m) (/ k_m t_m) 2.0)))
        (/ (* l_m l_m) (exp (fma (log t_m) 3.0 (* (log k_m) 2.0)))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 3.45e-137) {
		tmp = 2.0 / ((fma(fma(0.3333333333333333, pow(t_m, 3.0), t_m), (k_m * k_m), (2.0 * pow(t_m, 3.0))) / (l_m * l_m)) * (k_m * k_m));
	} else if (t_m <= 2.25e-69) {
		tmp = 2.0 / (((((k_m / t_m) * (k_m / t_m)) + 1.0) + 1.0) * ((((t_m * t_m) / l_m) * (t_m / l_m)) * (fma(fma(0.08611111111111111, (k_m * k_m), 0.16666666666666666), (k_m * k_m), 1.0) * (k_m * k_m))));
	} else if (t_m <= 1.05e+102) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * sin(k_m)) * tan(k_m)) * fma((k_m / t_m), (k_m / t_m), 2.0));
	} else {
		tmp = (l_m * l_m) / exp(fma(log(t_m), 3.0, (log(k_m) * 2.0)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 3.45e-137)
		tmp = Float64(2.0 / Float64(Float64(fma(fma(0.3333333333333333, (t_m ^ 3.0), t_m), Float64(k_m * k_m), Float64(2.0 * (t_m ^ 3.0))) / Float64(l_m * l_m)) * Float64(k_m * k_m)));
	elseif (t_m <= 2.25e-69)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k_m / t_m) * Float64(k_m / t_m)) + 1.0) + 1.0) * Float64(Float64(Float64(Float64(t_m * t_m) / l_m) * Float64(t_m / l_m)) * Float64(fma(fma(0.08611111111111111, Float64(k_m * k_m), 0.16666666666666666), Float64(k_m * k_m), 1.0) * Float64(k_m * k_m)))));
	elseif (t_m <= 1.05e+102)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / Float64(l_m * l_m)) * sin(k_m)) * tan(k_m)) * fma(Float64(k_m / t_m), Float64(k_m / t_m), 2.0)));
	else
		tmp = Float64(Float64(l_m * l_m) / exp(fma(log(t_m), 3.0, Float64(log(k_m) * 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.45e-137], N[(2.0 / N[(N[(N[(N[(0.3333333333333333 * N[Power[t$95$m, 3.0], $MachinePrecision] + t$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(2.0 * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.25e-69], N[(2.0 / N[(N[(N[(N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.08611111111111111 * N[(k$95$m * k$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.05e+102], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 3.44999999999999988e-137

   1. Initial program 51.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

   5. Applied rewrites 54.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, {t}^{3}, t\right), k \cdot k, 2 \cdot {t}^{3}\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   if 3.44999999999999988e-137 < t < 2.25000000000000005e-69

   1. Initial program 42.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left({k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) + 1\right) \cdot {\color{blue}{k}}^{2}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}, {k}^{2}, 1\right) \cdot {\color{blue}{k}}^{2}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{31}{360} \cdot {k}^{2} + \frac{1}{6}, {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, {k}^{2}, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      10. pow2 N/A

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)\right)} \]

      12. pow2 N/A

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)\right)} \]

      13. lift-*.f64 50.4

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)\right)} \]

   7. Applied rewrites 50.4%

      \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left(\frac{k}{t}\right)}}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t}} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      6. lift-/.f64 50.4

         \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \color{blue}{\frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

   9. Applied rewrites 50.4%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       3. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       4. associate-/r* N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       5. pow3 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       6. pow2 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\color{blue}{{t}^{2}} \cdot t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       7. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       9. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\color{blue}{\frac{{t}^{2}}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       10. pow2 N/A

           \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       11. lift-*.f64 N/A

           \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       12. lower-/.f64 58.7

           \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

   11. Applied rewrites 58.7%

       \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

   if 2.25000000000000005e-69 < t < 1.05000000000000001e102

   1. Initial program 86.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-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. unpow3 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{2}} \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{2} \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right)} \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-*.f64 86.8

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right)} \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 86.8%

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left({\color{blue}{\left(\frac{k}{t}\right)}}^{2} + 2\right)} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 2\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 2\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{k}{t}}, \frac{k}{t}, 2\right)} \]

      11. lift-/.f64 86.8

          \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \color{blue}{\frac{k}{t}}, 2\right)} \]

   6. Applied rewrites 86.8%

      \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   if 1.05000000000000001e102 < t

   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

   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

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 40.0

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 40.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      4. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{\ell \cdot \ell}{e^{\log t \cdot 3} \cdot {\color{blue}{k}}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{e^{3 \cdot \log t} \cdot {k}^{2}} \]

      8. pow-to-exp N/A

         \[\leadsto \frac{\ell \cdot \ell}{e^{3 \cdot \log t} \cdot e^{\log k \cdot 2}} \]

      9. prod-exp N/A

         \[\leadsto \frac{\ell \cdot \ell}{e^{3 \cdot \log t + \log k \cdot 2}} \]

      10. lower-exp.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{e^{3 \cdot \log t + \log k \cdot 2}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell \cdot \ell}{e^{\log t \cdot 3 + \log k \cdot 2}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]

      13. lift-log.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]

      15. lower-log.f64 29.5

          \[\leadsto \frac{\ell \cdot \ell}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]

   7. Applied rewrites 29.5%

      \[\leadsto \frac{\ell \cdot \ell}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 69.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.45 \cdot 10^{-137}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, {t\_m}^{3}, t\_m\right), k\_m \cdot k\_m, 2 \cdot {t\_m}^{3}\right)}{l\_m \cdot l\_m} \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{elif}\;t\_m \leq 2.25 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k\_m}{t\_m} \cdot \frac{k\_m}{t\_m} + 1\right) + 1\right) \cdot \left(\left(\frac{t\_m \cdot t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k\_m \cdot k\_m, 0.16666666666666666\right), k\_m \cdot k\_m, 1\right) \cdot \left(k\_m \cdot k\_m\right)\right)\right)}\\ \mathbf{elif}\;t\_m \leq 1.3 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{l\_m \cdot l\_m} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \mathsf{fma}\left(\frac{k\_m}{t\_m}, \frac{k\_m}{t\_m}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{{\left(k\_m \cdot t\_m\right)}^{2} \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.45e-137)
    (/
     2.0
     (*
      (/
       (fma
        (fma 0.3333333333333333 (pow t_m 3.0) t_m)
        (* k_m k_m)
        (* 2.0 (pow t_m 3.0)))
       (* l_m l_m))
      (* k_m k_m)))
    (if (<= t_m 2.25e-69)
      (/
       2.0
       (*
        (+ (+ (* (/ k_m t_m) (/ k_m t_m)) 1.0) 1.0)
        (*
         (* (/ (* t_m t_m) l_m) (/ t_m l_m))
         (*
          (fma
           (fma 0.08611111111111111 (* k_m k_m) 0.16666666666666666)
           (* k_m k_m)
           1.0)
          (* k_m k_m)))))
      (if (<= t_m 1.3e+102)
        (/
         2.0
         (*
          (* (* (/ (* (* t_m t_m) t_m) (* l_m l_m)) (sin k_m)) (tan k_m))
          (fma (/ k_m t_m) (/ k_m t_m) 2.0)))
        (/ (* l_m l_m) (* (pow (* k_m t_m) 2.0) t_m)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 3.45e-137) {
		tmp = 2.0 / ((fma(fma(0.3333333333333333, pow(t_m, 3.0), t_m), (k_m * k_m), (2.0 * pow(t_m, 3.0))) / (l_m * l_m)) * (k_m * k_m));
	} else if (t_m <= 2.25e-69) {
		tmp = 2.0 / (((((k_m / t_m) * (k_m / t_m)) + 1.0) + 1.0) * ((((t_m * t_m) / l_m) * (t_m / l_m)) * (fma(fma(0.08611111111111111, (k_m * k_m), 0.16666666666666666), (k_m * k_m), 1.0) * (k_m * k_m))));
	} else if (t_m <= 1.3e+102) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * sin(k_m)) * tan(k_m)) * fma((k_m / t_m), (k_m / t_m), 2.0));
	} else {
		tmp = (l_m * l_m) / (pow((k_m * t_m), 2.0) * t_m);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 3.45e-137)
		tmp = Float64(2.0 / Float64(Float64(fma(fma(0.3333333333333333, (t_m ^ 3.0), t_m), Float64(k_m * k_m), Float64(2.0 * (t_m ^ 3.0))) / Float64(l_m * l_m)) * Float64(k_m * k_m)));
	elseif (t_m <= 2.25e-69)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k_m / t_m) * Float64(k_m / t_m)) + 1.0) + 1.0) * Float64(Float64(Float64(Float64(t_m * t_m) / l_m) * Float64(t_m / l_m)) * Float64(fma(fma(0.08611111111111111, Float64(k_m * k_m), 0.16666666666666666), Float64(k_m * k_m), 1.0) * Float64(k_m * k_m)))));
	elseif (t_m <= 1.3e+102)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / Float64(l_m * l_m)) * sin(k_m)) * tan(k_m)) * fma(Float64(k_m / t_m), Float64(k_m / t_m), 2.0)));
	else
		tmp = Float64(Float64(l_m * l_m) / Float64((Float64(k_m * t_m) ^ 2.0) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.45e-137], N[(2.0 / N[(N[(N[(N[(0.3333333333333333 * N[Power[t$95$m, 3.0], $MachinePrecision] + t$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(2.0 * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.25e-69], N[(2.0 / N[(N[(N[(N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.08611111111111111 * N[(k$95$m * k$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.3e+102], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[Power[N[(k$95$m * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 3.44999999999999988e-137

   1. Initial program 51.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

   5. Applied rewrites 54.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, {t}^{3}, t\right), k \cdot k, 2 \cdot {t}^{3}\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   if 3.44999999999999988e-137 < t < 2.25000000000000005e-69

   1. Initial program 42.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left({k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) + 1\right) \cdot {\color{blue}{k}}^{2}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}, {k}^{2}, 1\right) \cdot {\color{blue}{k}}^{2}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{31}{360} \cdot {k}^{2} + \frac{1}{6}, {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, {k}^{2}, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      10. pow2 N/A

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)\right)} \]

      12. pow2 N/A

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)\right)} \]

      13. lift-*.f64 50.4

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)\right)} \]

   7. Applied rewrites 50.4%

      \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left(\frac{k}{t}\right)}}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t}} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      6. lift-/.f64 50.4

         \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \color{blue}{\frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

   9. Applied rewrites 50.4%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       3. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       4. associate-/r* N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       5. pow3 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       6. pow2 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\color{blue}{{t}^{2}} \cdot t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       7. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       9. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\color{blue}{\frac{{t}^{2}}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       10. pow2 N/A

           \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       11. lift-*.f64 N/A

           \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       12. lower-/.f64 58.7

           \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

   11. Applied rewrites 58.7%

       \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

   if 2.25000000000000005e-69 < t < 1.30000000000000003e102

   1. Initial program 86.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-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. unpow3 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{2}} \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{2} \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right)} \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-*.f64 86.8

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right)} \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 86.8%

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left({\color{blue}{\left(\frac{k}{t}\right)}}^{2} + 2\right)} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 2\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 2\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{k}{t}}, \frac{k}{t}, 2\right)} \]

      11. lift-/.f64 86.8

          \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \color{blue}{\frac{k}{t}}, 2\right)} \]

   6. Applied rewrites 86.8%

      \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   if 1.30000000000000003e102 < t

   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

   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

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 40.0

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 40.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 40.0

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   7. Applied rewrites 40.0%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      9. pow-prod-down N/A

         \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      11. lower-*.f64 66.9

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

   9. Applied rewrites 66.9%

      \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot \color{blue}{t}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 16: 69.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.45 \cdot 10^{-137}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, {t\_m}^{3}, t\_m\right), k\_m \cdot k\_m, 2 \cdot {t\_m}^{3}\right)}{l\_m \cdot l\_m} \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{elif}\;t\_m \leq 2.25 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k\_m}{t\_m} \cdot \frac{k\_m}{t\_m} + 1\right) + 1\right) \cdot \left(\left(\frac{t\_m \cdot t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k\_m \cdot k\_m, 0.16666666666666666\right), k\_m \cdot k\_m, 1\right) \cdot \left(k\_m \cdot k\_m\right)\right)\right)}\\ \mathbf{elif}\;t\_m \leq 8.4 \cdot 10^{+99}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{l\_m \cdot l\_m} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\frac{k\_m \cdot k\_m}{t\_m \cdot t\_m} + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{{\left(k\_m \cdot t\_m\right)}^{2} \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.45e-137)
    (/
     2.0
     (*
      (/
       (fma
        (fma 0.3333333333333333 (pow t_m 3.0) t_m)
        (* k_m k_m)
        (* 2.0 (pow t_m 3.0)))
       (* l_m l_m))
      (* k_m k_m)))
    (if (<= t_m 2.25e-69)
      (/
       2.0
       (*
        (+ (+ (* (/ k_m t_m) (/ k_m t_m)) 1.0) 1.0)
        (*
         (* (/ (* t_m t_m) l_m) (/ t_m l_m))
         (*
          (fma
           (fma 0.08611111111111111 (* k_m k_m) 0.16666666666666666)
           (* k_m k_m)
           1.0)
          (* k_m k_m)))))
      (if (<= t_m 8.4e+99)
        (/
         2.0
         (*
          (* (* (/ (* (* t_m t_m) t_m) (* l_m l_m)) (sin k_m)) (tan k_m))
          (+ (/ (* k_m k_m) (* t_m t_m)) 2.0)))
        (/ (* l_m l_m) (* (pow (* k_m t_m) 2.0) t_m)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 3.45e-137) {
		tmp = 2.0 / ((fma(fma(0.3333333333333333, pow(t_m, 3.0), t_m), (k_m * k_m), (2.0 * pow(t_m, 3.0))) / (l_m * l_m)) * (k_m * k_m));
	} else if (t_m <= 2.25e-69) {
		tmp = 2.0 / (((((k_m / t_m) * (k_m / t_m)) + 1.0) + 1.0) * ((((t_m * t_m) / l_m) * (t_m / l_m)) * (fma(fma(0.08611111111111111, (k_m * k_m), 0.16666666666666666), (k_m * k_m), 1.0) * (k_m * k_m))));
	} else if (t_m <= 8.4e+99) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * sin(k_m)) * tan(k_m)) * (((k_m * k_m) / (t_m * t_m)) + 2.0));
	} else {
		tmp = (l_m * l_m) / (pow((k_m * t_m), 2.0) * t_m);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 3.45e-137)
		tmp = Float64(2.0 / Float64(Float64(fma(fma(0.3333333333333333, (t_m ^ 3.0), t_m), Float64(k_m * k_m), Float64(2.0 * (t_m ^ 3.0))) / Float64(l_m * l_m)) * Float64(k_m * k_m)));
	elseif (t_m <= 2.25e-69)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k_m / t_m) * Float64(k_m / t_m)) + 1.0) + 1.0) * Float64(Float64(Float64(Float64(t_m * t_m) / l_m) * Float64(t_m / l_m)) * Float64(fma(fma(0.08611111111111111, Float64(k_m * k_m), 0.16666666666666666), Float64(k_m * k_m), 1.0) * Float64(k_m * k_m)))));
	elseif (t_m <= 8.4e+99)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / Float64(l_m * l_m)) * sin(k_m)) * tan(k_m)) * Float64(Float64(Float64(k_m * k_m) / Float64(t_m * t_m)) + 2.0)));
	else
		tmp = Float64(Float64(l_m * l_m) / Float64((Float64(k_m * t_m) ^ 2.0) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.45e-137], N[(2.0 / N[(N[(N[(N[(0.3333333333333333 * N[Power[t$95$m, 3.0], $MachinePrecision] + t$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(2.0 * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.25e-69], N[(2.0 / N[(N[(N[(N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.08611111111111111 * N[(k$95$m * k$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8.4e+99], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[Power[N[(k$95$m * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 3.44999999999999988e-137

   1. Initial program 51.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

   5. Applied rewrites 54.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, {t}^{3}, t\right), k \cdot k, 2 \cdot {t}^{3}\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   if 3.44999999999999988e-137 < t < 2.25000000000000005e-69

   1. Initial program 42.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left({k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) + 1\right) \cdot {\color{blue}{k}}^{2}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}, {k}^{2}, 1\right) \cdot {\color{blue}{k}}^{2}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{31}{360} \cdot {k}^{2} + \frac{1}{6}, {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, {k}^{2}, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      10. pow2 N/A

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)\right)} \]

      12. pow2 N/A

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)\right)} \]

      13. lift-*.f64 50.4

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)\right)} \]

   7. Applied rewrites 50.4%

      \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left(\frac{k}{t}\right)}}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t}} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      6. lift-/.f64 50.4

         \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \color{blue}{\frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

   9. Applied rewrites 50.4%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       3. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       4. associate-/r* N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       5. pow3 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       6. pow2 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\color{blue}{{t}^{2}} \cdot t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       7. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       9. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\color{blue}{\frac{{t}^{2}}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       10. pow2 N/A

           \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       11. lift-*.f64 N/A

           \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       12. lower-/.f64 58.7

           \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

   11. Applied rewrites 58.7%

       \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

   if 2.25000000000000005e-69 < t < 8.40000000000000041e99

   1. Initial program 86.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-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. unpow3 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{2}} \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{2} \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right)} \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-*.f64 86.4

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right)} \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 86.4%

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(2 + \frac{{k}^{2}}{{t}^{2}}\right)}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(2 + \frac{{k}^{2}}{{t}^{2}}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(2 + \frac{{k}^{2}}{{t}^{2}}\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(2 + \frac{{k}^{2}}{{t}^{2}}\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(2 + \frac{{k}^{2}}{{t}^{2}}\right)} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(2 + \frac{{k}^{2}}{{t}^{2}}\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\frac{{k}^{2}}{{t}^{2}} + \color{blue}{2}\right)} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\frac{{k}^{2}}{{t}^{2}} + \color{blue}{2}\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\frac{{k}^{2}}{{t}^{2}} + 2\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\frac{k \cdot k}{{t}^{2}} + 2\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\frac{k \cdot k}{{t}^{2}} + 2\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\frac{k \cdot k}{t \cdot t} + 2\right)} \]

      12. lift-*.f64 83.2

          \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\frac{k \cdot k}{t \cdot t} + 2\right)} \]

   7. Applied rewrites 83.2%

      \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(\frac{k \cdot k}{t \cdot t} + 2\right)}} \]

   if 8.40000000000000041e99 < t

   1. Initial program 53.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 39.1

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 39.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 39.1

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   7. Applied rewrites 39.1%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      9. pow-prod-down N/A

         \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      11. lower-*.f64 65.1

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

   9. Applied rewrites 65.1%

      \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot \color{blue}{t}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 17: 65.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.66 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{t\_m}^{3}}{l\_m}}{l\_m} \cdot k\_m\right) \cdot \left(\tan k\_m \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, {t\_m}^{3}, t\_m\right), k\_m \cdot k\_m, 2 \cdot {t\_m}^{3}\right)}{l\_m \cdot l\_m} \cdot \left(k\_m \cdot k\_m\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.66e-29)
    (/ 2.0 (* (* (/ (/ (pow t_m 3.0) l_m) l_m) k_m) (* (tan k_m) 2.0)))
    (/
     2.0
     (*
      (/
       (fma
        (fma 0.3333333333333333 (pow t_m 3.0) t_m)
        (* k_m k_m)
        (* 2.0 (pow t_m 3.0)))
       (* l_m l_m))
      (* k_m k_m))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 1.66e-29) {
		tmp = 2.0 / ((((pow(t_m, 3.0) / l_m) / l_m) * k_m) * (tan(k_m) * 2.0));
	} else {
		tmp = 2.0 / ((fma(fma(0.3333333333333333, pow(t_m, 3.0), t_m), (k_m * k_m), (2.0 * pow(t_m, 3.0))) / (l_m * l_m)) * (k_m * k_m));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 1.66e-29)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / l_m) / l_m) * k_m) * Float64(tan(k_m) * 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(fma(fma(0.3333333333333333, (t_m ^ 3.0), t_m), Float64(k_m * k_m), Float64(2.0 * (t_m ^ 3.0))) / Float64(l_m * l_m)) * Float64(k_m * k_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.66e-29], N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(0.3333333333333333 * N[Power[t$95$m, 3.0], $MachinePrecision] + t$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(2.0 * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.6600000000000001e-29

   1. Initial program 57.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{2}} \]
   4. Step-by-step derivation

   5. Applied rewrites 56.9%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{2}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{k}\right) \cdot \tan k\right) \cdot 2} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.0%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{k}\right) \cdot \tan k\right) \cdot 2} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot k\right) \cdot \tan k\right) \cdot 2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot k\right) \cdot \tan k\right)} \cdot 2} \]

      3. lift-tan.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot k\right) \cdot \color{blue}{\tan k}\right) \cdot 2} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot k\right) \cdot \left(\tan k \cdot 2\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot k\right) \cdot \left(\tan k \cdot 2\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot k\right) \cdot \left(\tan k \cdot 2\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot k\right) \cdot \left(\tan k \cdot 2\right)} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot k\right) \cdot \left(\tan k \cdot 2\right)} \]

      9. associate-/r* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot k\right) \cdot \left(\tan k \cdot 2\right)} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell} \cdot k\right) \cdot \left(\tan k \cdot 2\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell} \cdot k\right) \cdot \left(\tan k \cdot 2\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot k\right) \cdot \left(\tan k \cdot 2\right)} \]

   10. Applied rewrites 62.2%

       \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot k\right) \cdot \left(\tan k \cdot 2\right)}} \]

   if 1.6600000000000001e-29 < k

   1. Initial program 48.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

   5. Applied rewrites 59.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, {t}^{3}, t\right), k \cdot k, 2 \cdot {t}^{3}\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.66 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot k\right) \cdot \left(\tan k \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, {t}^{3}, t\right), k \cdot k, 2 \cdot {t}^{3}\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 65.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.2 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{t\_m}^{3}}{l\_m}}{l\_m} \cdot k\_m\right) \cdot \left(\tan k\_m \cdot 2\right)}\\ \mathbf{elif}\;t\_m \leq 2.9 \cdot 10^{+22}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k\_m}{t\_m} \cdot \frac{k\_m}{t\_m} + 1\right) + 1\right) \cdot \left(\left(\frac{t\_m \cdot t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k\_m \cdot k\_m, 0.16666666666666666\right), k\_m \cdot k\_m, 1\right) \cdot \left(k\_m \cdot k\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{{\left(k\_m \cdot t\_m\right)}^{2} \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.2e-161)
    (/ 2.0 (* (* (/ (/ (pow t_m 3.0) l_m) l_m) k_m) (* (tan k_m) 2.0)))
    (if (<= t_m 2.9e+22)
      (/
       2.0
       (*
        (+ (+ (* (/ k_m t_m) (/ k_m t_m)) 1.0) 1.0)
        (*
         (* (/ (* t_m t_m) l_m) (/ t_m l_m))
         (*
          (fma
           (fma 0.08611111111111111 (* k_m k_m) 0.16666666666666666)
           (* k_m k_m)
           1.0)
          (* k_m k_m)))))
      (/ (* l_m l_m) (* (pow (* k_m t_m) 2.0) t_m))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 6.2e-161) {
		tmp = 2.0 / ((((pow(t_m, 3.0) / l_m) / l_m) * k_m) * (tan(k_m) * 2.0));
	} else if (t_m <= 2.9e+22) {
		tmp = 2.0 / (((((k_m / t_m) * (k_m / t_m)) + 1.0) + 1.0) * ((((t_m * t_m) / l_m) * (t_m / l_m)) * (fma(fma(0.08611111111111111, (k_m * k_m), 0.16666666666666666), (k_m * k_m), 1.0) * (k_m * k_m))));
	} else {
		tmp = (l_m * l_m) / (pow((k_m * t_m), 2.0) * t_m);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 6.2e-161)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / l_m) / l_m) * k_m) * Float64(tan(k_m) * 2.0)));
	elseif (t_m <= 2.9e+22)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k_m / t_m) * Float64(k_m / t_m)) + 1.0) + 1.0) * Float64(Float64(Float64(Float64(t_m * t_m) / l_m) * Float64(t_m / l_m)) * Float64(fma(fma(0.08611111111111111, Float64(k_m * k_m), 0.16666666666666666), Float64(k_m * k_m), 1.0) * Float64(k_m * k_m)))));
	else
		tmp = Float64(Float64(l_m * l_m) / Float64((Float64(k_m * t_m) ^ 2.0) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 6.2e-161], N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.9e+22], N[(2.0 / N[(N[(N[(N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.08611111111111111 * N[(k$95$m * k$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[Power[N[(k$95$m * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 6.1999999999999997e-161

   1. Initial program 52.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{2}} \]
   4. Step-by-step derivation

   5. Applied rewrites 53.4%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{2}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{k}\right) \cdot \tan k\right) \cdot 2} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.4%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{k}\right) \cdot \tan k\right) \cdot 2} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot k\right) \cdot \tan k\right) \cdot 2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot k\right) \cdot \tan k\right)} \cdot 2} \]

      3. lift-tan.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot k\right) \cdot \color{blue}{\tan k}\right) \cdot 2} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot k\right) \cdot \left(\tan k \cdot 2\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot k\right) \cdot \left(\tan k \cdot 2\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot k\right) \cdot \left(\tan k \cdot 2\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot k\right) \cdot \left(\tan k \cdot 2\right)} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot k\right) \cdot \left(\tan k \cdot 2\right)} \]

      9. associate-/r* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot k\right) \cdot \left(\tan k \cdot 2\right)} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell} \cdot k\right) \cdot \left(\tan k \cdot 2\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell} \cdot k\right) \cdot \left(\tan k \cdot 2\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot k\right) \cdot \left(\tan k \cdot 2\right)} \]

   10. Applied rewrites 59.1%

       \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot k\right) \cdot \left(\tan k \cdot 2\right)}} \]

   if 6.1999999999999997e-161 < t < 2.9e22

   1. Initial program 55.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 59.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left({k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) + 1\right) \cdot {\color{blue}{k}}^{2}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}, {k}^{2}, 1\right) \cdot {\color{blue}{k}}^{2}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{31}{360} \cdot {k}^{2} + \frac{1}{6}, {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, {k}^{2}, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      10. pow2 N/A

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)\right)} \]

      12. pow2 N/A

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)\right)} \]

      13. lift-*.f64 55.7

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)\right)} \]

   7. Applied rewrites 55.7%

      \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left(\frac{k}{t}\right)}}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t}} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      6. lift-/.f64 55.7

         \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \color{blue}{\frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

   9. Applied rewrites 55.7%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       3. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       4. associate-/r* N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       5. pow3 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       6. pow2 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\color{blue}{{t}^{2}} \cdot t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       7. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       9. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\color{blue}{\frac{{t}^{2}}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       10. pow2 N/A

           \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       11. lift-*.f64 N/A

           \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       12. lower-/.f64 62.6

           \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

   11. Applied rewrites 62.6%

       \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

   if 2.9e22 < t

   1. Initial program 64.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 44.8

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 44.8%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 44.8

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   7. Applied rewrites 44.8%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      9. pow-prod-down N/A

         \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      11. lower-*.f64 65.4

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

   9. Applied rewrites 65.4%

      \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot \color{blue}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.2 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot k\right) \cdot \left(\tan k \cdot 2\right)}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+22}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 63.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.4 \cdot 10^{-161} \lor \neg \left(t\_m \leq 7.5 \cdot 10^{-40}\right):\\ \;\;\;\;l\_m \cdot \frac{l\_m}{k\_m \cdot \left({t\_m}^{3} \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k\_m}{t\_m} \cdot \frac{k\_m}{t\_m} + 1\right) + 1\right) \cdot \left(\left(\frac{t\_m \cdot t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k\_m \cdot k\_m, 0.16666666666666666\right), k\_m \cdot k\_m, 1\right) \cdot \left(k\_m \cdot k\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (or (<= t_m 7.4e-161) (not (<= t_m 7.5e-40)))
    (* l_m (/ l_m (* k_m (* (pow t_m 3.0) k_m))))
    (/
     2.0
     (*
      (+ (+ (* (/ k_m t_m) (/ k_m t_m)) 1.0) 1.0)
      (*
       (* (/ (* t_m t_m) l_m) (/ t_m l_m))
       (*
        (fma
         (fma 0.08611111111111111 (* k_m k_m) 0.16666666666666666)
         (* k_m k_m)
         1.0)
        (* k_m k_m))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if ((t_m <= 7.4e-161) || !(t_m <= 7.5e-40)) {
		tmp = l_m * (l_m / (k_m * (pow(t_m, 3.0) * k_m)));
	} else {
		tmp = 2.0 / (((((k_m / t_m) * (k_m / t_m)) + 1.0) + 1.0) * ((((t_m * t_m) / l_m) * (t_m / l_m)) * (fma(fma(0.08611111111111111, (k_m * k_m), 0.16666666666666666), (k_m * k_m), 1.0) * (k_m * k_m))));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if ((t_m <= 7.4e-161) || !(t_m <= 7.5e-40))
		tmp = Float64(l_m * Float64(l_m / Float64(k_m * Float64((t_m ^ 3.0) * k_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k_m / t_m) * Float64(k_m / t_m)) + 1.0) + 1.0) * Float64(Float64(Float64(Float64(t_m * t_m) / l_m) * Float64(t_m / l_m)) * Float64(fma(fma(0.08611111111111111, Float64(k_m * k_m), 0.16666666666666666), Float64(k_m * k_m), 1.0) * Float64(k_m * k_m)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[Or[LessEqual[t$95$m, 7.4e-161], N[Not[LessEqual[t$95$m, 7.5e-40]], $MachinePrecision]], N[(l$95$m * N[(l$95$m / N[(k$95$m * N[(N[Power[t$95$m, 3.0], $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.08611111111111111 * N[(k$95$m * k$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.3999999999999995e-161 or 7.50000000000000069e-40 < t

   1. Initial program 56.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 47.0

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 47.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      3. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      8. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      9. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      13. lift-*.f64 53.0

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

   7. Applied rewrites 53.0%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      4. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      6. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left({t}^{3} \cdot \color{blue}{k}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left({t}^{3} \cdot \color{blue}{k}\right)} \]

      8. lift-pow.f64 59.4

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)} \]

   9. Applied rewrites 59.4%

      \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left({t}^{3} \cdot k\right)}} \]

   if 7.3999999999999995e-161 < t < 7.50000000000000069e-40

   1. Initial program 43.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 47.9%

      \[\leadsto \color{blue}{\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left({k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) + 1\right) \cdot {\color{blue}{k}}^{2}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}, {k}^{2}, 1\right) \cdot {\color{blue}{k}}^{2}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{31}{360} \cdot {k}^{2} + \frac{1}{6}, {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, {k}^{2}, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      10. pow2 N/A

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)\right)} \]

      12. pow2 N/A

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)\right)} \]

      13. lift-*.f64 47.8

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)\right)} \]

   7. Applied rewrites 47.8%

      \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left(\frac{k}{t}\right)}}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t}} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      6. lift-/.f64 47.8

         \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \color{blue}{\frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

   9. Applied rewrites 47.8%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       3. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       4. associate-/r* N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       5. pow3 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       6. pow2 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\color{blue}{{t}^{2}} \cdot t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       7. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       9. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\color{blue}{\frac{{t}^{2}}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       10. pow2 N/A

           \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       11. lift-*.f64 N/A

           \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       12. lower-/.f64 57.3

           \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

   11. Applied rewrites 57.3%

       \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.4 \cdot 10^{-161} \lor \neg \left(t \leq 7.5 \cdot 10^{-40}\right):\\ \;\;\;\;\ell \cdot \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 65.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.4 \cdot 10^{-161}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{k\_m \cdot \left({t\_m}^{3} \cdot k\_m\right)}\\ \mathbf{elif}\;t\_m \leq 2.9 \cdot 10^{+22}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k\_m}{t\_m} \cdot \frac{k\_m}{t\_m} + 1\right) + 1\right) \cdot \left(\left(\frac{t\_m \cdot t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k\_m \cdot k\_m, 0.16666666666666666\right), k\_m \cdot k\_m, 1\right) \cdot \left(k\_m \cdot k\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{{\left(k\_m \cdot t\_m\right)}^{2} \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.4e-161)
    (* l_m (/ l_m (* k_m (* (pow t_m 3.0) k_m))))
    (if (<= t_m 2.9e+22)
      (/
       2.0
       (*
        (+ (+ (* (/ k_m t_m) (/ k_m t_m)) 1.0) 1.0)
        (*
         (* (/ (* t_m t_m) l_m) (/ t_m l_m))
         (*
          (fma
           (fma 0.08611111111111111 (* k_m k_m) 0.16666666666666666)
           (* k_m k_m)
           1.0)
          (* k_m k_m)))))
      (/ (* l_m l_m) (* (pow (* k_m t_m) 2.0) t_m))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 7.4e-161) {
		tmp = l_m * (l_m / (k_m * (pow(t_m, 3.0) * k_m)));
	} else if (t_m <= 2.9e+22) {
		tmp = 2.0 / (((((k_m / t_m) * (k_m / t_m)) + 1.0) + 1.0) * ((((t_m * t_m) / l_m) * (t_m / l_m)) * (fma(fma(0.08611111111111111, (k_m * k_m), 0.16666666666666666), (k_m * k_m), 1.0) * (k_m * k_m))));
	} else {
		tmp = (l_m * l_m) / (pow((k_m * t_m), 2.0) * t_m);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 7.4e-161)
		tmp = Float64(l_m * Float64(l_m / Float64(k_m * Float64((t_m ^ 3.0) * k_m))));
	elseif (t_m <= 2.9e+22)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k_m / t_m) * Float64(k_m / t_m)) + 1.0) + 1.0) * Float64(Float64(Float64(Float64(t_m * t_m) / l_m) * Float64(t_m / l_m)) * Float64(fma(fma(0.08611111111111111, Float64(k_m * k_m), 0.16666666666666666), Float64(k_m * k_m), 1.0) * Float64(k_m * k_m)))));
	else
		tmp = Float64(Float64(l_m * l_m) / Float64((Float64(k_m * t_m) ^ 2.0) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 7.4e-161], N[(l$95$m * N[(l$95$m / N[(k$95$m * N[(N[Power[t$95$m, 3.0], $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.9e+22], N[(2.0 / N[(N[(N[(N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.08611111111111111 * N[(k$95$m * k$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[Power[N[(k$95$m * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 7.3999999999999995e-161

   1. Initial program 52.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 46.3

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 46.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      3. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      8. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      9. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      13. lift-*.f64 53.1

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

   7. Applied rewrites 53.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      4. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      6. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left({t}^{3} \cdot \color{blue}{k}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left({t}^{3} \cdot \color{blue}{k}\right)} \]

      8. lift-pow.f64 58.3

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)} \]

   9. Applied rewrites 58.3%

      \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left({t}^{3} \cdot k\right)}} \]

   if 7.3999999999999995e-161 < t < 2.9e22

   1. Initial program 55.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 59.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left({k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) + 1\right) \cdot {\color{blue}{k}}^{2}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}, {k}^{2}, 1\right) \cdot {\color{blue}{k}}^{2}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{31}{360} \cdot {k}^{2} + \frac{1}{6}, {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, {k}^{2}, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      10. pow2 N/A

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)\right)} \]

      12. pow2 N/A

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)\right)} \]

      13. lift-*.f64 55.7

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)\right)} \]

   7. Applied rewrites 55.7%

      \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left(\frac{k}{t}\right)}}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t}} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      6. lift-/.f64 55.7

         \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \color{blue}{\frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

   9. Applied rewrites 55.7%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       3. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       4. associate-/r* N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       5. pow3 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       6. pow2 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\color{blue}{{t}^{2}} \cdot t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       7. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       9. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\color{blue}{\frac{{t}^{2}}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       10. pow2 N/A

           \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       11. lift-*.f64 N/A

           \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       12. lower-/.f64 62.6

           \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

   11. Applied rewrites 62.6%

       \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

   if 2.9e22 < t

   1. Initial program 64.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 44.8

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 44.8%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 44.8

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   7. Applied rewrites 44.8%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      9. pow-prod-down N/A

         \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      11. lower-*.f64 65.4

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

   9. Applied rewrites 65.4%

      \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot \color{blue}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 59.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.2 \cdot 10^{-161}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{\left(k\_m \cdot k\_m\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k\_m}{t\_m} \cdot \frac{k\_m}{t\_m} + 1\right) + 1\right) \cdot \left(\left(\frac{t\_m \cdot t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k\_m \cdot k\_m, 0.16666666666666666\right), k\_m \cdot k\_m, 1\right) \cdot \left(k\_m \cdot k\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.2e-161)
    (* l_m (/ l_m (* (* k_m k_m) (* (* t_m t_m) t_m))))
    (/
     2.0
     (*
      (+ (+ (* (/ k_m t_m) (/ k_m t_m)) 1.0) 1.0)
      (*
       (* (/ (* t_m t_m) l_m) (/ t_m l_m))
       (*
        (fma
         (fma 0.08611111111111111 (* k_m k_m) 0.16666666666666666)
         (* k_m k_m)
         1.0)
        (* k_m k_m))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 6.2e-161) {
		tmp = l_m * (l_m / ((k_m * k_m) * ((t_m * t_m) * t_m)));
	} else {
		tmp = 2.0 / (((((k_m / t_m) * (k_m / t_m)) + 1.0) + 1.0) * ((((t_m * t_m) / l_m) * (t_m / l_m)) * (fma(fma(0.08611111111111111, (k_m * k_m), 0.16666666666666666), (k_m * k_m), 1.0) * (k_m * k_m))));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 6.2e-161)
		tmp = Float64(l_m * Float64(l_m / Float64(Float64(k_m * k_m) * Float64(Float64(t_m * t_m) * t_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k_m / t_m) * Float64(k_m / t_m)) + 1.0) + 1.0) * Float64(Float64(Float64(Float64(t_m * t_m) / l_m) * Float64(t_m / l_m)) * Float64(fma(fma(0.08611111111111111, Float64(k_m * k_m), 0.16666666666666666), Float64(k_m * k_m), 1.0) * Float64(k_m * k_m)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 6.2e-161], N[(l$95$m * N[(l$95$m / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.08611111111111111 * N[(k$95$m * k$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.1999999999999997e-161

   1. Initial program 52.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 46.3

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 46.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      3. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      8. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      9. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      13. lift-*.f64 53.1

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

   7. Applied rewrites 53.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. pow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 53.1

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   9. Applied rewrites 53.1%

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   if 6.1999999999999997e-161 < t

   1. Initial program 61.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 58.5%

      \[\leadsto \color{blue}{\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left({k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) + 1\right) \cdot {\color{blue}{k}}^{2}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}, {k}^{2}, 1\right) \cdot {\color{blue}{k}}^{2}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{31}{360} \cdot {k}^{2} + \frac{1}{6}, {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, {k}^{2}, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)\right)} \]

      10. pow2 N/A

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)\right)} \]

      12. pow2 N/A

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)\right)} \]

      13. lift-*.f64 50.4

          \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)\right)} \]

   7. Applied rewrites 50.4%

      \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left(\frac{k}{t}\right)}}^{2} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t}} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

      6. lift-/.f64 50.4

         \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \color{blue}{\frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

   9. Applied rewrites 50.4%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       3. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       4. associate-/r* N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       5. pow3 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       6. pow2 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\frac{\color{blue}{{t}^{2}} \cdot t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       7. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       9. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\color{blue}{\frac{{t}^{2}}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       10. pow2 N/A

           \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       11. lift-*.f64 N/A

           \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

       12. lower-/.f64 55.4

           \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]

   11. Applied rewrites 55.4%

       \[\leadsto \frac{2}{\left(\left(\frac{k}{t} \cdot \frac{k}{t} + 1\right) + 1\right) \cdot \left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 55.9% accurate, 12.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(l\_m \cdot \frac{l\_m}{\left(k\_m \cdot k\_m\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (* t_s (* l_m (/ l_m (* (* k_m k_m) (* (* t_m t_m) t_m))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	return t_s * (l_m * (l_m / ((k_m * k_m) * ((t_m * t_m) * t_m))));
}

Fortran

l_m =     private
k_m =     private
t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    code = t_s * (l_m * (l_m / ((k_m * k_m) * ((t_m * t_m) * t_m))))
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	return t_s * (l_m * (l_m / ((k_m * k_m) * ((t_m * t_m) * t_m))));
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	return t_s * (l_m * (l_m / ((k_m * k_m) * ((t_m * t_m) * t_m))))

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	return Float64(t_s * Float64(l_m * Float64(l_m / Float64(Float64(k_m * k_m) * Float64(Float64(t_m * t_m) * t_m)))))
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k_m)
	tmp = t_s * (l_m * (l_m / ((k_m * k_m) * ((t_m * t_m) * t_m))));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * N[(l$95$m * N[(l$95$m / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   7. lift-pow.f64 46.4

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

5. Applied rewrites 46.4%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   3. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   5. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

   6. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   7. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   8. pow2 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   9. lower-/.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   10. pow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   11. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   12. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   13. lift-*.f64 52.3

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

7. Applied rewrites 52.3%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
8. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   2. pow3 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   4. lift-*.f64 52.3

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

9. Applied rewrites 52.3%

   \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025072 
(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))))