Toniolo and Linder, Equation (10+)

Percentage Accurate: 56.0% → 89.1%

Time: 12.3s

Alternatives: 25

Speedup: 10.7×

• 27.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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: 56.0% 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: 89.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{2}{\frac{t\_m}{\ell}}\\ t_3 := \frac{\tan k \cdot t\_m}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.3 \cdot 10^{-78}:\\ \;\;\;\;\frac{t\_2}{\frac{\left({\sin k}^{2} \cdot k\right) \cdot k}{\cos k \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m} \cdot t\_3, t\_3 \cdot 2\right) \cdot \left(\sin k \cdot t\_m\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ 2.0 (/ t_m l))) (t_3 (/ (* (tan k) t_m) l)))
   (*
    t_s
    (if (<= t_m 1.3e-78)
      (/ t_2 (/ (* (* (pow (sin k) 2.0) k) k) (* (cos k) l)))
      (/
       t_2
       (* (fma (/ k t_m) (* (/ k t_m) t_3) (* t_3 2.0)) (* (sin k) t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 / (t_m / l);
	double t_3 = (tan(k) * t_m) / l;
	double tmp;
	if (t_m <= 1.3e-78) {
		tmp = t_2 / (((pow(sin(k), 2.0) * k) * k) / (cos(k) * l));
	} else {
		tmp = t_2 / (fma((k / t_m), ((k / t_m) * t_3), (t_3 * 2.0)) * (sin(k) * t_m));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(2.0 / Float64(t_m / l))
	t_3 = Float64(Float64(tan(k) * t_m) / l)
	tmp = 0.0
	if (t_m <= 1.3e-78)
		tmp = Float64(t_2 / Float64(Float64(Float64((sin(k) ^ 2.0) * k) * k) / Float64(cos(k) * l)));
	else
		tmp = Float64(t_2 / Float64(fma(Float64(k / t_m), Float64(Float64(k / t_m) * t_3), Float64(t_3 * 2.0)) * Float64(sin(k) * t_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.3e-78], N[(t$95$2 / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * t$95$3), $MachinePrecision] + N[(t$95$3 * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.3000000000000001e-78

   1. Initial program 48.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 72.3

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

   4. Applied rewrites 72.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 76.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 79.0

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

   8. Applied rewrites 79.0%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-pow.f64 N/A

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

       8. lower-sin.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lower-cos.f64 71.5

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

   11. Applied rewrites 71.5%

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

   if 1.3000000000000001e-78 < t

   1. Initial program 67.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 86.8

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

   4. Applied rewrites 86.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 92.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 89.6

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

   8. Applied rewrites 89.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft-in N/A

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

   10. Applied rewrites 89.7%

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

Alternative 2: 68.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 10^{+306}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k} \cdot \frac{\ell}{k \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t\_m} \cdot \ell}{t\_m}}{t\_m \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       1e+306)
    (* (/ (/ l (* t_m t_m)) k) (/ l (* k t_m)))
    (/ (/ (* (/ l t_m) l) t_m) (* t_m (* k k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= 1e+306) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	} else {
		tmp = (((l / t_m) * l) / t_m) / (t_m * (k * k));
	}
	return t_s * tmp;
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))) <= 1d+306) then
        tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m))
    else
        tmp = (((l / t_m) * l) / t_m) / (t_m * (k * k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((2.0 / ((((Math.pow(t_m, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0))) <= 1e+306) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	} else {
		tmp = (((l / t_m) * l) / t_m) / (t_m * (k * k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (2.0 / ((((math.pow(t_m, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))) <= 1e+306:
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m))
	else:
		tmp = (((l / t_m) * l) / t_m) / (t_m * (k * k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))) <= 1e+306)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / k) * Float64(l / Float64(k * t_m)));
	else
		tmp = Float64(Float64(Float64(Float64(l / t_m) * l) / t_m) / Float64(t_m * Float64(k * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((2.0 / (((((t_m ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0))) <= 1e+306)
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	else
		tmp = (((l / t_m) * l) / t_m) / (t_m * (k * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+306], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 1.00000000000000002e306

   1. Initial program 78.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 67.0%

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

   9. Applied rewrites 67.1%

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

   11. Applied rewrites 78.5%

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

   if 1.00000000000000002e306 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 22.7%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 37.7

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

   5. Applied rewrites 37.7%

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

   7. Applied rewrites 37.7%

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

   9. Applied rewrites 41.4%

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

   11. Applied rewrites 52.7%

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

Alternative 3: 68.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 10^{+306}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k} \cdot \frac{\ell}{k \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m}}{t\_m \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       1e+306)
    (* (/ (/ l (* t_m t_m)) k) (/ l (* k t_m)))
    (/ (* (/ l t_m) (/ l t_m)) (* t_m (* k k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= 1e+306) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	} else {
		tmp = ((l / t_m) * (l / t_m)) / (t_m * (k * k));
	}
	return t_s * tmp;
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))) <= 1d+306) then
        tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m))
    else
        tmp = ((l / t_m) * (l / t_m)) / (t_m * (k * k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((2.0 / ((((Math.pow(t_m, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0))) <= 1e+306) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	} else {
		tmp = ((l / t_m) * (l / t_m)) / (t_m * (k * k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (2.0 / ((((math.pow(t_m, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))) <= 1e+306:
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m))
	else:
		tmp = ((l / t_m) * (l / t_m)) / (t_m * (k * k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))) <= 1e+306)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / k) * Float64(l / Float64(k * t_m)));
	else
		tmp = Float64(Float64(Float64(l / t_m) * Float64(l / t_m)) / Float64(t_m * Float64(k * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((2.0 / (((((t_m ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0))) <= 1e+306)
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	else
		tmp = ((l / t_m) * (l / t_m)) / (t_m * (k * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+306], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 1.00000000000000002e306

   1. Initial program 78.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 67.0%

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

   9. Applied rewrites 67.1%

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

   11. Applied rewrites 78.5%

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

   if 1.00000000000000002e306 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 22.7%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 37.7

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

   5. Applied rewrites 37.7%

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

   7. Applied rewrites 37.7%

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

   9. Applied rewrites 41.4%

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

   11. Applied rewrites 52.0%

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

Alternative 4: 87.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sin k \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.3 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell}}}{\frac{\left({\sin k}^{2} \cdot k\right) \cdot k}{\cos k \cdot \ell}}\\ \mathbf{elif}\;t\_m \leq 1.1 \cdot 10^{+189}:\\ \;\;\;\;\frac{\frac{2}{t\_2 \cdot t\_m} \cdot \ell}{\left(t\_m \cdot \left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right)\right) \cdot \tan k} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \sin k\right)\right) \cdot \left(\frac{2}{\cos k} \cdot \frac{t\_2}{\ell}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (sin k) t_m)))
   (*
    t_s
    (if (<= t_m 1.3e-78)
      (/ (/ 2.0 (/ t_m l)) (/ (* (* (pow (sin k) 2.0) k) k) (* (cos k) l)))
      (if (<= t_m 1.1e+189)
        (*
         (/
          (* (/ 2.0 (* t_2 t_m)) l)
          (* (* t_m (+ (pow (/ k t_m) 2.0) 2.0)) (tan k)))
         l)
        (/
         2.0
         (* (* (/ t_m l) (* t_m (sin k))) (* (/ 2.0 (cos k)) (/ t_2 l)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = sin(k) * t_m;
	double tmp;
	if (t_m <= 1.3e-78) {
		tmp = (2.0 / (t_m / l)) / (((pow(sin(k), 2.0) * k) * k) / (cos(k) * l));
	} else if (t_m <= 1.1e+189) {
		tmp = (((2.0 / (t_2 * t_m)) * l) / ((t_m * (pow((k / t_m), 2.0) + 2.0)) * tan(k))) * l;
	} else {
		tmp = 2.0 / (((t_m / l) * (t_m * sin(k))) * ((2.0 / cos(k)) * (t_2 / l)));
	}
	return t_s * tmp;
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sin(k) * t_m
    if (t_m <= 1.3d-78) then
        tmp = (2.0d0 / (t_m / l)) / ((((sin(k) ** 2.0d0) * k) * k) / (cos(k) * l))
    else if (t_m <= 1.1d+189) then
        tmp = (((2.0d0 / (t_2 * t_m)) * l) / ((t_m * (((k / t_m) ** 2.0d0) + 2.0d0)) * tan(k))) * l
    else
        tmp = 2.0d0 / (((t_m / l) * (t_m * sin(k))) * ((2.0d0 / cos(k)) * (t_2 / l)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.sin(k) * t_m;
	double tmp;
	if (t_m <= 1.3e-78) {
		tmp = (2.0 / (t_m / l)) / (((Math.pow(Math.sin(k), 2.0) * k) * k) / (Math.cos(k) * l));
	} else if (t_m <= 1.1e+189) {
		tmp = (((2.0 / (t_2 * t_m)) * l) / ((t_m * (Math.pow((k / t_m), 2.0) + 2.0)) * Math.tan(k))) * l;
	} else {
		tmp = 2.0 / (((t_m / l) * (t_m * Math.sin(k))) * ((2.0 / Math.cos(k)) * (t_2 / l)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.sin(k) * t_m
	tmp = 0
	if t_m <= 1.3e-78:
		tmp = (2.0 / (t_m / l)) / (((math.pow(math.sin(k), 2.0) * k) * k) / (math.cos(k) * l))
	elif t_m <= 1.1e+189:
		tmp = (((2.0 / (t_2 * t_m)) * l) / ((t_m * (math.pow((k / t_m), 2.0) + 2.0)) * math.tan(k))) * l
	else:
		tmp = 2.0 / (((t_m / l) * (t_m * math.sin(k))) * ((2.0 / math.cos(k)) * (t_2 / l)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(sin(k) * t_m)
	tmp = 0.0
	if (t_m <= 1.3e-78)
		tmp = Float64(Float64(2.0 / Float64(t_m / l)) / Float64(Float64(Float64((sin(k) ^ 2.0) * k) * k) / Float64(cos(k) * l)));
	elseif (t_m <= 1.1e+189)
		tmp = Float64(Float64(Float64(Float64(2.0 / Float64(t_2 * t_m)) * l) / Float64(Float64(t_m * Float64((Float64(k / t_m) ^ 2.0) + 2.0)) * tan(k))) * l);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(t_m * sin(k))) * Float64(Float64(2.0 / cos(k)) * Float64(t_2 / l))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = sin(k) * t_m;
	tmp = 0.0;
	if (t_m <= 1.3e-78)
		tmp = (2.0 / (t_m / l)) / ((((sin(k) ^ 2.0) * k) * k) / (cos(k) * l));
	elseif (t_m <= 1.1e+189)
		tmp = (((2.0 / (t_2 * t_m)) * l) / ((t_m * (((k / t_m) ^ 2.0) + 2.0)) * tan(k))) * l;
	else
		tmp = 2.0 / (((t_m / l) * (t_m * sin(k))) * ((2.0 / cos(k)) * (t_2 / l)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.3e-78], N[(N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.1e+189], N[(N[(N[(N[(2.0 / N[(t$95$2 * t$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / N[(N[(t$95$m * N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.3000000000000001e-78

   1. Initial program 48.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 72.3

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

   4. Applied rewrites 72.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 76.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 79.0

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

   8. Applied rewrites 79.0%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-pow.f64 N/A

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

       8. lower-sin.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lower-cos.f64 71.5

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

   11. Applied rewrites 71.5%

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

   if 1.3000000000000001e-78 < t < 1.10000000000000003e189

   1. Initial program 72.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 84.7

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

   4. Applied rewrites 84.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 90.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

   8. Applied rewrites 83.4%

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

   if 1.10000000000000003e189 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 93.9

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

   4. Applied rewrites 93.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 99.6

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

   9. Applied rewrites 99.6%

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

Alternative 5: 89.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.26 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell}}}{\frac{\left({\sin k}^{2} \cdot k\right) \cdot k}{\cos k \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \sin k\right)\right) \cdot \left(\frac{t\_m}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.26e-78)
    (/ (/ 2.0 (/ t_m l)) (/ (* (* (pow (sin k) 2.0) k) k) (* (cos k) l)))
    (/
     2.0
     (*
      (* (/ t_m l) (* t_m (sin k)))
      (* (/ t_m l) (* (tan k) (+ (pow (/ k t_m) 2.0) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.26e-78) {
		tmp = (2.0 / (t_m / l)) / (((pow(sin(k), 2.0) * k) * k) / (cos(k) * l));
	} else {
		tmp = 2.0 / (((t_m / l) * (t_m * sin(k))) * ((t_m / l) * (tan(k) * (pow((k / t_m), 2.0) + 2.0))));
	}
	return t_s * tmp;
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.26d-78) then
        tmp = (2.0d0 / (t_m / l)) / ((((sin(k) ** 2.0d0) * k) * k) / (cos(k) * l))
    else
        tmp = 2.0d0 / (((t_m / l) * (t_m * sin(k))) * ((t_m / l) * (tan(k) * (((k / t_m) ** 2.0d0) + 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.26e-78) {
		tmp = (2.0 / (t_m / l)) / (((Math.pow(Math.sin(k), 2.0) * k) * k) / (Math.cos(k) * l));
	} else {
		tmp = 2.0 / (((t_m / l) * (t_m * Math.sin(k))) * ((t_m / l) * (Math.tan(k) * (Math.pow((k / t_m), 2.0) + 2.0))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.26e-78:
		tmp = (2.0 / (t_m / l)) / (((math.pow(math.sin(k), 2.0) * k) * k) / (math.cos(k) * l))
	else:
		tmp = 2.0 / (((t_m / l) * (t_m * math.sin(k))) * ((t_m / l) * (math.tan(k) * (math.pow((k / t_m), 2.0) + 2.0))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.26e-78)
		tmp = Float64(Float64(2.0 / Float64(t_m / l)) / Float64(Float64(Float64((sin(k) ^ 2.0) * k) * k) / Float64(cos(k) * l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(t_m * sin(k))) * Float64(Float64(t_m / l) * Float64(tan(k) * Float64((Float64(k / t_m) ^ 2.0) + 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.26e-78)
		tmp = (2.0 / (t_m / l)) / ((((sin(k) ^ 2.0) * k) * k) / (cos(k) * l));
	else
		tmp = 2.0 / (((t_m / l) * (t_m * sin(k))) * ((t_m / l) * (tan(k) * (((k / t_m) ^ 2.0) + 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.26e-78], N[(N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.26000000000000008e-78

   1. Initial program 48.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 72.3

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

   4. Applied rewrites 72.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 76.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 79.0

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

   8. Applied rewrites 79.0%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-pow.f64 N/A

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

       8. lower-sin.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lower-cos.f64 71.5

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

   11. Applied rewrites 71.5%

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

   if 1.26000000000000008e-78 < t

   1. Initial program 67.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 86.8

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

   4. Applied rewrites 86.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 92.8%

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

Alternative 6: 85.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.26 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell}}}{\frac{\left({\sin k}^{2} \cdot k\right) \cdot k}{\cos k \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right) \cdot \left(\left(\left(\tan k \cdot t\_m\right) \cdot \sin k\right) \cdot \frac{t\_m}{\ell}\right)\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.26e-78)
    (/ (/ 2.0 (/ t_m l)) (/ (* (* (pow (sin k) 2.0) k) k) (* (cos k) l)))
    (/
     2.0
     (*
      (* (+ (pow (/ k t_m) 2.0) 2.0) (* (* (* (tan k) t_m) (sin k)) (/ t_m l)))
      (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.26e-78) {
		tmp = (2.0 / (t_m / l)) / (((pow(sin(k), 2.0) * k) * k) / (cos(k) * l));
	} else {
		tmp = 2.0 / (((pow((k / t_m), 2.0) + 2.0) * (((tan(k) * t_m) * sin(k)) * (t_m / l))) * (t_m / l));
	}
	return t_s * tmp;
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.26d-78) then
        tmp = (2.0d0 / (t_m / l)) / ((((sin(k) ** 2.0d0) * k) * k) / (cos(k) * l))
    else
        tmp = 2.0d0 / (((((k / t_m) ** 2.0d0) + 2.0d0) * (((tan(k) * t_m) * sin(k)) * (t_m / l))) * (t_m / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.26e-78) {
		tmp = (2.0 / (t_m / l)) / (((Math.pow(Math.sin(k), 2.0) * k) * k) / (Math.cos(k) * l));
	} else {
		tmp = 2.0 / (((Math.pow((k / t_m), 2.0) + 2.0) * (((Math.tan(k) * t_m) * Math.sin(k)) * (t_m / l))) * (t_m / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.26e-78:
		tmp = (2.0 / (t_m / l)) / (((math.pow(math.sin(k), 2.0) * k) * k) / (math.cos(k) * l))
	else:
		tmp = 2.0 / (((math.pow((k / t_m), 2.0) + 2.0) * (((math.tan(k) * t_m) * math.sin(k)) * (t_m / l))) * (t_m / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.26e-78)
		tmp = Float64(Float64(2.0 / Float64(t_m / l)) / Float64(Float64(Float64((sin(k) ^ 2.0) * k) * k) / Float64(cos(k) * l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 2.0) * Float64(Float64(Float64(tan(k) * t_m) * sin(k)) * Float64(t_m / l))) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.26e-78)
		tmp = (2.0 / (t_m / l)) / ((((sin(k) ^ 2.0) * k) * k) / (cos(k) * l));
	else
		tmp = 2.0 / (((((k / t_m) ^ 2.0) + 2.0) * (((tan(k) * t_m) * sin(k)) * (t_m / l))) * (t_m / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.26e-78], N[(N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.26000000000000008e-78

   1. Initial program 48.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 72.3

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

   4. Applied rewrites 72.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 76.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 79.0

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

   8. Applied rewrites 79.0%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-pow.f64 N/A

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

       8. lower-sin.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lower-cos.f64 71.5

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

   11. Applied rewrites 71.5%

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

   if 1.26000000000000008e-78 < t

   1. Initial program 67.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 86.8

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

   4. Applied rewrites 86.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

   6. Applied rewrites 87.0%

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

Alternative 7: 84.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.3 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell}}}{\frac{\left({\sin k}^{2} \cdot k\right) \cdot k}{\cos k \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\sin k \cdot t\_m\right) \cdot t\_m} \cdot \ell}{\left(t\_m \cdot \left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right)\right) \cdot \tan k} \cdot \ell\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.3e-78)
    (/ (/ 2.0 (/ t_m l)) (/ (* (* (pow (sin k) 2.0) k) k) (* (cos k) l)))
    (*
     (/
      (* (/ 2.0 (* (* (sin k) t_m) t_m)) l)
      (* (* t_m (+ (pow (/ k t_m) 2.0) 2.0)) (tan k)))
     l))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.3e-78) {
		tmp = (2.0 / (t_m / l)) / (((pow(sin(k), 2.0) * k) * k) / (cos(k) * l));
	} else {
		tmp = (((2.0 / ((sin(k) * t_m) * t_m)) * l) / ((t_m * (pow((k / t_m), 2.0) + 2.0)) * tan(k))) * l;
	}
	return t_s * tmp;
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.3d-78) then
        tmp = (2.0d0 / (t_m / l)) / ((((sin(k) ** 2.0d0) * k) * k) / (cos(k) * l))
    else
        tmp = (((2.0d0 / ((sin(k) * t_m) * t_m)) * l) / ((t_m * (((k / t_m) ** 2.0d0) + 2.0d0)) * tan(k))) * l
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.3e-78) {
		tmp = (2.0 / (t_m / l)) / (((Math.pow(Math.sin(k), 2.0) * k) * k) / (Math.cos(k) * l));
	} else {
		tmp = (((2.0 / ((Math.sin(k) * t_m) * t_m)) * l) / ((t_m * (Math.pow((k / t_m), 2.0) + 2.0)) * Math.tan(k))) * l;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.3e-78:
		tmp = (2.0 / (t_m / l)) / (((math.pow(math.sin(k), 2.0) * k) * k) / (math.cos(k) * l))
	else:
		tmp = (((2.0 / ((math.sin(k) * t_m) * t_m)) * l) / ((t_m * (math.pow((k / t_m), 2.0) + 2.0)) * math.tan(k))) * l
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.3e-78)
		tmp = Float64(Float64(2.0 / Float64(t_m / l)) / Float64(Float64(Float64((sin(k) ^ 2.0) * k) * k) / Float64(cos(k) * l)));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 / Float64(Float64(sin(k) * t_m) * t_m)) * l) / Float64(Float64(t_m * Float64((Float64(k / t_m) ^ 2.0) + 2.0)) * tan(k))) * l);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.3e-78)
		tmp = (2.0 / (t_m / l)) / ((((sin(k) ^ 2.0) * k) * k) / (cos(k) * l));
	else
		tmp = (((2.0 / ((sin(k) * t_m) * t_m)) * l) / ((t_m * (((k / t_m) ^ 2.0) + 2.0)) * tan(k))) * l;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.3e-78], N[(N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / N[(N[(t$95$m * N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.3000000000000001e-78

   1. Initial program 48.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 72.3

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

   4. Applied rewrites 72.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 76.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 79.0

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

   8. Applied rewrites 79.0%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-pow.f64 N/A

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

       8. lower-sin.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lower-cos.f64 71.5

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

   11. Applied rewrites 71.5%

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

   if 1.3000000000000001e-78 < t

   1. Initial program 67.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 86.8

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

   4. Applied rewrites 86.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 92.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

   8. Applied rewrites 83.3%

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

Alternative 8: 81.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2} + 2\\ t_3 := \frac{2}{\frac{t\_m}{\ell}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.52 \cdot 10^{-78}:\\ \;\;\;\;\frac{t\_3}{\frac{\left({\sin k}^{2} \cdot k\right) \cdot k}{\cos k \cdot \ell}}\\ \mathbf{elif}\;t\_m \leq 1.25 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{t\_m \cdot t\_m}{\ell}\right) \cdot \left(\frac{t\_m}{\ell} \cdot \left(\tan k \cdot t\_2\right)\right)}\\ \mathbf{elif}\;t\_m \leq 3.5 \cdot 10^{+163}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\tan k \cdot \sin k\right) \cdot t\_m\right) \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right)\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{\left(\left(\frac{t\_m}{\ell} \cdot k\right) \cdot t\_2\right) \cdot \left(\sin k \cdot t\_m\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (+ (pow (/ k t_m) 2.0) 2.0)) (t_3 (/ 2.0 (/ t_m l))))
   (*
    t_s
    (if (<= t_m 1.52e-78)
      (/ t_3 (/ (* (* (pow (sin k) 2.0) k) k) (* (cos k) l)))
      (if (<= t_m 1.25e+16)
        (/ 2.0 (* (* k (/ (* t_m t_m) l)) (* (/ t_m l) (* (tan k) t_2))))
        (if (<= t_m 3.5e+163)
          (/
           2.0
           (*
            (* (* (* (tan k) (sin k)) t_m) (* (/ t_m l) (/ t_m l)))
            (fma (/ k t_m) (/ k t_m) 2.0)))
          (/ t_3 (* (* (* (/ t_m l) k) t_2) (* (sin k) t_m)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0) + 2.0;
	double t_3 = 2.0 / (t_m / l);
	double tmp;
	if (t_m <= 1.52e-78) {
		tmp = t_3 / (((pow(sin(k), 2.0) * k) * k) / (cos(k) * l));
	} else if (t_m <= 1.25e+16) {
		tmp = 2.0 / ((k * ((t_m * t_m) / l)) * ((t_m / l) * (tan(k) * t_2)));
	} else if (t_m <= 3.5e+163) {
		tmp = 2.0 / ((((tan(k) * sin(k)) * t_m) * ((t_m / l) * (t_m / l))) * fma((k / t_m), (k / t_m), 2.0));
	} else {
		tmp = t_3 / ((((t_m / l) * k) * t_2) * (sin(k) * t_m));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64((Float64(k / t_m) ^ 2.0) + 2.0)
	t_3 = Float64(2.0 / Float64(t_m / l))
	tmp = 0.0
	if (t_m <= 1.52e-78)
		tmp = Float64(t_3 / Float64(Float64(Float64((sin(k) ^ 2.0) * k) * k) / Float64(cos(k) * l)));
	elseif (t_m <= 1.25e+16)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(Float64(t_m * t_m) / l)) * Float64(Float64(t_m / l) * Float64(tan(k) * t_2))));
	elseif (t_m <= 3.5e+163)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k) * sin(k)) * t_m) * Float64(Float64(t_m / l) * Float64(t_m / l))) * fma(Float64(k / t_m), Float64(k / t_m), 2.0)));
	else
		tmp = Float64(t_3 / Float64(Float64(Float64(Float64(t_m / l) * k) * t_2) * Float64(sin(k) * t_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.52e-78], N[(t$95$3 / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.25e+16], N[(2.0 / N[(N[(k * N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.5e+163], N[(2.0 / N[(N[(N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.52e-78

   1. Initial program 48.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 72.3

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

   4. Applied rewrites 72.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 76.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 79.0

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

   8. Applied rewrites 79.0%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-pow.f64 N/A

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

       8. lower-sin.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lower-cos.f64 71.5

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

   11. Applied rewrites 71.5%

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

   if 1.52e-78 < t < 1.25e16

   1. Initial program 76.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 76.7

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

   4. Applied rewrites 76.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 88.6%

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

   7. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 86.0

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

   9. Applied rewrites 86.0%

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

   if 1.25e16 < t < 3.5000000000000003e163

   1. Initial program 68.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. cube-mult N/A

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

      10. associate-/l* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. times-frac N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-/.f64 85.2

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

   4. Applied rewrites 85.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 85.2

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

   6. Applied rewrites 85.2%

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

   if 3.5000000000000003e163 < t

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 95.1

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

   4. Applied rewrites 95.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.6

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

   8. Applied rewrites 99.5%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 91.1

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

   11. Applied rewrites 91.1%

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

Alternative 9: 76.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-64}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \sin k\right)\right) \cdot \left(\frac{t\_m}{\ell} \cdot \left(2 \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 11000:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell}}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot k, k, \left(2 \cdot t\_m\right) \cdot t\_m\right)}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(\left({\sin k}^{2} \cdot t\_m\right) \cdot k\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2e-64)
    (/ 2.0 (* (* (/ t_m l) (* t_m (sin k))) (* (/ t_m l) (* 2.0 k))))
    (if (<= k 11000.0)
      (/
       (/ 2.0 (/ t_m l))
       (*
        (/
         (fma
          (* (fma 0.3333333333333333 (* t_m t_m) 1.0) k)
          k
          (* (* 2.0 t_m) t_m))
         l)
        (* k k)))
      (* (* 2.0 (* l l)) (/ (cos k) (* (* (* (pow (sin k) 2.0) t_m) k) k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2e-64) {
		tmp = 2.0 / (((t_m / l) * (t_m * sin(k))) * ((t_m / l) * (2.0 * k)));
	} else if (k <= 11000.0) {
		tmp = (2.0 / (t_m / l)) / ((fma((fma(0.3333333333333333, (t_m * t_m), 1.0) * k), k, ((2.0 * t_m) * t_m)) / l) * (k * k));
	} else {
		tmp = (2.0 * (l * l)) * (cos(k) / (((pow(sin(k), 2.0) * t_m) * k) * k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2e-64)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(t_m * sin(k))) * Float64(Float64(t_m / l) * Float64(2.0 * k))));
	elseif (k <= 11000.0)
		tmp = Float64(Float64(2.0 / Float64(t_m / l)) / Float64(Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * k), k, Float64(Float64(2.0 * t_m) * t_m)) / l) * Float64(k * k)));
	else
		tmp = Float64(Float64(2.0 * Float64(l * l)) * Float64(cos(k) / Float64(Float64(Float64((sin(k) ^ 2.0) * t_m) * k) * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2e-64], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 11000.0], N[(N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.99999999999999993e-64

   1. Initial program 56.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 80.7

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

   4. Applied rewrites 80.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 86.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-*.f64 76.6

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

   9. Applied rewrites 76.6%

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

   if 1.99999999999999993e-64 < k < 11000

   1. Initial program 69.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 81.5

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

   4. Applied rewrites 81.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 81.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)\right)}} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{t}{\ell}}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right) \cdot \left(t \cdot \sin k\right)}} \]

      9. lower-*.f64 87.6

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right) \cdot \left(t \cdot \sin k\right)}} \]

   8. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(\frac{\tan k \cdot t}{\ell} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(\sin k \cdot t\right)}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}}} \]

   11. Applied rewrites 100.0%

       \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot k, k, \left(2 \cdot t\right) \cdot t\right)}{\ell} \cdot \left(k \cdot k\right)}} \]

   if 11000 < k

   1. Initial program 44.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 65.1

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 65.1%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. unpow2 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      8. lower-cos.f64 N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\color{blue}{\cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\color{blue}{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      11. associate-*r* N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\color{blue}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\color{blue}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot k}} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\color{blue}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)} \cdot k} \]

      14. *-commutative N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(\color{blue}{\left({\sin k}^{2} \cdot t\right)} \cdot k\right) \cdot k} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(\color{blue}{\left({\sin k}^{2} \cdot t\right)} \cdot k\right) \cdot k} \]

      16. lower-pow.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(\left(\color{blue}{{\sin k}^{2}} \cdot t\right) \cdot k\right) \cdot k} \]

      17. lower-sin.f64 70.2

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(\left({\color{blue}{\sin k}}^{2} \cdot t\right) \cdot k\right) \cdot k} \]

   7. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(\left({\sin k}^{2} \cdot t\right) \cdot k\right) \cdot k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-64}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(2 \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 11000:\\ \;\;\;\;\frac{\frac{2}{\frac{t}{\ell}}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot k, k, \left(2 \cdot t\right) \cdot t\right)}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(\left({\sin k}^{2} \cdot t\right) \cdot k\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-64}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \sin k\right)\right) \cdot \left(\frac{t\_m}{\ell} \cdot \left(2 \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 3.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell}}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot k, k, \left(2 \cdot t\_m\right) \cdot t\_m\right)}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\tan k \cdot \sin k\right) \cdot t\_m\right) \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right)\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2e-64)
    (/ 2.0 (* (* (/ t_m l) (* t_m (sin k))) (* (/ t_m l) (* 2.0 k))))
    (if (<= k 3.2e-19)
      (/
       (/ 2.0 (/ t_m l))
       (*
        (/
         (fma
          (* (fma 0.3333333333333333 (* t_m t_m) 1.0) k)
          k
          (* (* 2.0 t_m) t_m))
         l)
        (* k k)))
      (/
       2.0
       (*
        (* (* (* (tan k) (sin k)) t_m) (* (/ t_m l) (/ t_m l)))
        (fma (/ k t_m) (/ k t_m) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2e-64) {
		tmp = 2.0 / (((t_m / l) * (t_m * sin(k))) * ((t_m / l) * (2.0 * k)));
	} else if (k <= 3.2e-19) {
		tmp = (2.0 / (t_m / l)) / ((fma((fma(0.3333333333333333, (t_m * t_m), 1.0) * k), k, ((2.0 * t_m) * t_m)) / l) * (k * k));
	} else {
		tmp = 2.0 / ((((tan(k) * sin(k)) * t_m) * ((t_m / l) * (t_m / l))) * fma((k / t_m), (k / t_m), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2e-64)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(t_m * sin(k))) * Float64(Float64(t_m / l) * Float64(2.0 * k))));
	elseif (k <= 3.2e-19)
		tmp = Float64(Float64(2.0 / Float64(t_m / l)) / Float64(Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * k), k, Float64(Float64(2.0 * t_m) * t_m)) / l) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k) * sin(k)) * t_m) * Float64(Float64(t_m / l) * Float64(t_m / l))) * fma(Float64(k / t_m), Float64(k / t_m), 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2e-64], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.2e-19], N[(N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.99999999999999993e-64

   1. Initial program 56.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 80.7

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 80.7%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\sin k \cdot t\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

   6. Applied rewrites 86.9%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(2 \cdot k\right)}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 76.6

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(2 \cdot k\right)}\right)} \]

   9. Applied rewrites 76.6%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(2 \cdot k\right)}\right)} \]

   if 1.99999999999999993e-64 < k < 3.19999999999999982e-19

   1. Initial program 44.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 67.1

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 67.1%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\sin k \cdot t\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

   6. Applied rewrites 67.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)\right)}} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{t}{\ell}}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right) \cdot \left(t \cdot \sin k\right)}} \]

      9. lower-*.f64 78.0

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right) \cdot \left(t \cdot \sin k\right)}} \]

   8. Applied rewrites 78.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(\frac{\tan k \cdot t}{\ell} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(\sin k \cdot t\right)}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}}} \]

   11. Applied rewrites 100.0%

       \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot k, k, \left(2 \cdot t\right) \cdot t\right)}{\ell} \cdot \left(k \cdot k\right)}} \]

   if 3.19999999999999982e-19 < k

   1. Initial program 49.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\tan k \cdot \sin k\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. associate-/l* N/A

          \[\leadsto \frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. associate-*r* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot t\right)} \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\tan k \cdot \sin k\right)} \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\tan k \cdot \sin k\right)} \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\tan k \cdot \sin k\right) \cdot t\right) \cdot \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      17. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\left(\tan k \cdot \sin k\right) \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\tan k \cdot \sin k\right) \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      19. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\tan k \cdot \sin k\right) \cdot t\right) \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      20. lower-/.f64 68.2

          \[\leadsto \frac{2}{\left(\left(\left(\tan k \cdot \sin k\right) \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 68.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\tan k \cdot \sin k\right) \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\tan k \cdot \sin k\right) \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\tan k \cdot \sin k\right) \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\left(\tan k \cdot \sin k\right) \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2}{\left(\left(\left(\tan k \cdot \sin k\right) \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left(\left(\tan k \cdot \sin k\right) \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\tan k \cdot \sin k\right) \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 2\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\tan k \cdot \sin k\right) \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 2\right)} \]

      8. lower-fma.f64 68.2

         \[\leadsto \frac{2}{\left(\left(\left(\tan k \cdot \sin k\right) \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   6. Applied rewrites 68.2%

      \[\leadsto \frac{2}{\left(\left(\left(\tan k \cdot \sin k\right) \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 77.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell}}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot k, k, \left(2 \cdot t\_m\right) \cdot t\_m\right)}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \sin k\right)\right) \cdot \left(\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot t\_m, 0.6666666666666666 + {\left(t\_m \cdot t\_m\right)}^{-1}, 2 \cdot t\_m\right)}{\ell} \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8e-116)
    (/
     (/ 2.0 (/ t_m l))
     (*
      (/
       (fma
        (* (fma 0.3333333333333333 (* t_m t_m) 1.0) k)
        k
        (* (* 2.0 t_m) t_m))
       l)
      (* k k)))
    (/
     2.0
     (*
      (* (/ t_m l) (* t_m (sin k)))
      (*
       (/
        (fma
         (* (* k k) t_m)
         (+ 0.6666666666666666 (pow (* t_m t_m) -1.0))
         (* 2.0 t_m))
        l)
       k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8e-116) {
		tmp = (2.0 / (t_m / l)) / ((fma((fma(0.3333333333333333, (t_m * t_m), 1.0) * k), k, ((2.0 * t_m) * t_m)) / l) * (k * k));
	} else {
		tmp = 2.0 / (((t_m / l) * (t_m * sin(k))) * ((fma(((k * k) * t_m), (0.6666666666666666 + pow((t_m * t_m), -1.0)), (2.0 * t_m)) / l) * k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8e-116)
		tmp = Float64(Float64(2.0 / Float64(t_m / l)) / Float64(Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * k), k, Float64(Float64(2.0 * t_m) * t_m)) / l) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(t_m * sin(k))) * Float64(Float64(fma(Float64(Float64(k * k) * t_m), Float64(0.6666666666666666 + (Float64(t_m * t_m) ^ -1.0)), Float64(2.0 * t_m)) / l) * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8e-116], N[(N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(0.6666666666666666 + N[Power[N[(t$95$m * t$95$m), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8e-116

   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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 72.3

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 72.3%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\sin k \cdot t\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

   6. Applied rewrites 77.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)\right)}} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{t}{\ell}}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right) \cdot \left(t \cdot \sin k\right)}} \]

      9. lower-*.f64 79.2

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right) \cdot \left(t \cdot \sin k\right)}} \]

   8. Applied rewrites 79.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(\frac{\tan k \cdot t}{\ell} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(\sin k \cdot t\right)}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}}} \]

   11. Applied rewrites 64.7%

       \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot k, k, \left(2 \cdot t\right) \cdot t\right)}{\ell} \cdot \left(k \cdot k\right)}} \]

   if 8e-116 < t

   1. Initial program 67.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 85.5

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 85.5%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\sin k \cdot t\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

   6. Applied rewrites 91.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \left(2 \cdot \frac{t}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(\left(2 \cdot \frac{t}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot k\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(\left(2 \cdot \frac{t}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot k\right)}} \]

   9. Applied rewrites 81.7%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, 0.6666666666666666 + \frac{1}{t \cdot t}, 2 \cdot t\right)}{\ell} \cdot k\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{2}{\frac{t}{\ell}}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot k, k, \left(2 \cdot t\right) \cdot t\right)}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, 0.6666666666666666 + {\left(t \cdot t\right)}^{-1}, 2 \cdot t\right)}{\ell} \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sin k \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 4.8 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell}}}{\left(\left(\frac{t\_m}{\ell} \cdot k\right) \cdot \left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right)\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t\_2 \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (sin k) t_m)))
   (*
    t_s
    (if (<= l 4.8e+86)
      (/
       (/ 2.0 (/ t_m l))
       (* (* (* (/ t_m l) k) (+ (pow (/ k t_m) 2.0) 2.0)) t_2))
      (/ 2.0 (* (* (* t_2 (* (/ t_m l) (/ t_m l))) (tan k)) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = sin(k) * t_m;
	double tmp;
	if (l <= 4.8e+86) {
		tmp = (2.0 / (t_m / l)) / ((((t_m / l) * k) * (pow((k / t_m), 2.0) + 2.0)) * t_2);
	} else {
		tmp = 2.0 / (((t_2 * ((t_m / l) * (t_m / l))) * tan(k)) * 2.0);
	}
	return t_s * tmp;
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sin(k) * t_m
    if (l <= 4.8d+86) then
        tmp = (2.0d0 / (t_m / l)) / ((((t_m / l) * k) * (((k / t_m) ** 2.0d0) + 2.0d0)) * t_2)
    else
        tmp = 2.0d0 / (((t_2 * ((t_m / l) * (t_m / l))) * tan(k)) * 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.sin(k) * t_m;
	double tmp;
	if (l <= 4.8e+86) {
		tmp = (2.0 / (t_m / l)) / ((((t_m / l) * k) * (Math.pow((k / t_m), 2.0) + 2.0)) * t_2);
	} else {
		tmp = 2.0 / (((t_2 * ((t_m / l) * (t_m / l))) * Math.tan(k)) * 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.sin(k) * t_m
	tmp = 0
	if l <= 4.8e+86:
		tmp = (2.0 / (t_m / l)) / ((((t_m / l) * k) * (math.pow((k / t_m), 2.0) + 2.0)) * t_2)
	else:
		tmp = 2.0 / (((t_2 * ((t_m / l) * (t_m / l))) * math.tan(k)) * 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(sin(k) * t_m)
	tmp = 0.0
	if (l <= 4.8e+86)
		tmp = Float64(Float64(2.0 / Float64(t_m / l)) / Float64(Float64(Float64(Float64(t_m / l) * k) * Float64((Float64(k / t_m) ^ 2.0) + 2.0)) * t_2));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * Float64(Float64(t_m / l) * Float64(t_m / l))) * tan(k)) * 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = sin(k) * t_m;
	tmp = 0.0;
	if (l <= 4.8e+86)
		tmp = (2.0 / (t_m / l)) / ((((t_m / l) * k) * (((k / t_m) ^ 2.0) + 2.0)) * t_2);
	else
		tmp = 2.0 / (((t_2 * ((t_m / l) * (t_m / l))) * tan(k)) * 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l, 4.8e+86], N[(N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision] * N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$2 * N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.8000000000000001e86

   1. Initial program 59.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 79.2

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 79.2%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\sin k \cdot t\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

   6. Applied rewrites 83.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)\right)}} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{t}{\ell}}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right) \cdot \left(t \cdot \sin k\right)}} \]

      9. lower-*.f64 84.6

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right) \cdot \left(t \cdot \sin k\right)}} \]

   8. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(\frac{\tan k \cdot t}{\ell} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(\sin k \cdot t\right)}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\left(\color{blue}{\frac{k \cdot t}{\ell}} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(\sin k \cdot t\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\left(\frac{\color{blue}{t \cdot k}}{\ell} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(\sin k \cdot t\right)} \]

       2. associate-*l/ N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot k\right)} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(\sin k \cdot t\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot k\right)} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(\sin k \cdot t\right)} \]

       4. lower-/.f64 78.9

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\left(\left(\color{blue}{\frac{t}{\ell}} \cdot k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(\sin k \cdot t\right)} \]

   11. Applied rewrites 78.9%

       \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot k\right)} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(\sin k \cdot t\right)} \]

   if 4.8000000000000001e86 < l

   1. Initial program 28.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 63.0

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 63.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \color{blue}{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 64.7%

      \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \color{blue}{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 77.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.7 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell}}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot k, k, \left(2 \cdot t\_m\right) \cdot t\_m\right)}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \sin k\right)\right) \cdot \left(\frac{t\_m}{\ell} \cdot \left(\mathsf{fma}\left(0.6666666666666666 + {\left(t\_m \cdot t\_m\right)}^{-1}, k \cdot k, 2\right) \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.7e-105)
    (/
     (/ 2.0 (/ t_m l))
     (*
      (/
       (fma
        (* (fma 0.3333333333333333 (* t_m t_m) 1.0) k)
        k
        (* (* 2.0 t_m) t_m))
       l)
      (* k k)))
    (/
     2.0
     (*
      (* (/ t_m l) (* t_m (sin k)))
      (*
       (/ t_m l)
       (*
        (fma (+ 0.6666666666666666 (pow (* t_m t_m) -1.0)) (* k k) 2.0)
        k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.7e-105) {
		tmp = (2.0 / (t_m / l)) / ((fma((fma(0.3333333333333333, (t_m * t_m), 1.0) * k), k, ((2.0 * t_m) * t_m)) / l) * (k * k));
	} else {
		tmp = 2.0 / (((t_m / l) * (t_m * sin(k))) * ((t_m / l) * (fma((0.6666666666666666 + pow((t_m * t_m), -1.0)), (k * k), 2.0) * k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.7e-105)
		tmp = Float64(Float64(2.0 / Float64(t_m / l)) / Float64(Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * k), k, Float64(Float64(2.0 * t_m) * t_m)) / l) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(t_m * sin(k))) * Float64(Float64(t_m / l) * Float64(fma(Float64(0.6666666666666666 + (Float64(t_m * t_m) ^ -1.0)), Float64(k * k), 2.0) * k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.7e-105], N[(N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(0.6666666666666666 + N[Power[N[(t$95$m * t$95$m), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision] + 2.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.69999999999999963e-105

   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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 72.6

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 72.6%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\sin k \cdot t\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

   6. Applied rewrites 77.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)\right)}} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{t}{\ell}}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right) \cdot \left(t \cdot \sin k\right)}} \]

      9. lower-*.f64 79.5

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right) \cdot \left(t \cdot \sin k\right)}} \]

   8. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(\frac{\tan k \cdot t}{\ell} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(\sin k \cdot t\right)}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}}} \]

   11. Applied rewrites 65.1%

       \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot k, k, \left(2 \cdot t\right) \cdot t\right)}{\ell} \cdot \left(k \cdot k\right)}} \]

   if 5.69999999999999963e-105 < 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

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 85.2

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 85.2%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\sin k \cdot t\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

   6. Applied rewrites 90.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(k \cdot \left(2 + {k}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\left(2 + {k}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right) \cdot k\right)}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\left(2 + {k}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right) \cdot k\right)}\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\color{blue}{\left({k}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + 2\right)} \cdot k\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) \cdot {k}^{2}} + 2\right) \cdot k\right)\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{2}{3} + \frac{1}{{t}^{2}}, {k}^{2}, 2\right)} \cdot k\right)\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{2}{3} + \frac{1}{{t}^{2}}}, {k}^{2}, 2\right) \cdot k\right)\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\mathsf{fma}\left(\frac{2}{3} + \color{blue}{\frac{1}{{t}^{2}}}, {k}^{2}, 2\right) \cdot k\right)\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\mathsf{fma}\left(\frac{2}{3} + \frac{1}{\color{blue}{t \cdot t}}, {k}^{2}, 2\right) \cdot k\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\mathsf{fma}\left(\frac{2}{3} + \frac{1}{\color{blue}{t \cdot t}}, {k}^{2}, 2\right) \cdot k\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\mathsf{fma}\left(\frac{2}{3} + \frac{1}{t \cdot t}, \color{blue}{k \cdot k}, 2\right) \cdot k\right)\right)} \]

      11. lower-*.f64 81.3

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\mathsf{fma}\left(0.6666666666666666 + \frac{1}{t \cdot t}, \color{blue}{k \cdot k}, 2\right) \cdot k\right)\right)} \]

   9. Applied rewrites 81.3%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(0.6666666666666666 + \frac{1}{t \cdot t}, k \cdot k, 2\right) \cdot k\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.7 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{2}{\frac{t}{\ell}}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot k, k, \left(2 \cdot t\right) \cdot t\right)}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\mathsf{fma}\left(0.6666666666666666 + {\left(t \cdot t\right)}^{-1}, k \cdot k, 2\right) \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 72.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell}}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot k, k, \left(2 \cdot t\_m\right) \cdot t\_m\right)}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m \cdot t\_m}{\ell} \cdot k\right) \cdot \left(\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot t\_m, 0.6666666666666666 + {\left(t\_m \cdot t\_m\right)}^{-1}, 2 \cdot t\_m\right)}{\ell} \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.8e-78)
    (/
     (/ 2.0 (/ t_m l))
     (*
      (/
       (fma
        (* (fma 0.3333333333333333 (* t_m t_m) 1.0) k)
        k
        (* (* 2.0 t_m) t_m))
       l)
      (* k k)))
    (/
     2.0
     (*
      (* (/ (* t_m t_m) l) k)
      (*
       (/
        (fma
         (* (* k k) t_m)
         (+ 0.6666666666666666 (pow (* t_m t_m) -1.0))
         (* 2.0 t_m))
        l)
       k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.8e-78) {
		tmp = (2.0 / (t_m / l)) / ((fma((fma(0.3333333333333333, (t_m * t_m), 1.0) * k), k, ((2.0 * t_m) * t_m)) / l) * (k * k));
	} else {
		tmp = 2.0 / ((((t_m * t_m) / l) * k) * ((fma(((k * k) * t_m), (0.6666666666666666 + pow((t_m * t_m), -1.0)), (2.0 * t_m)) / l) * k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.8e-78)
		tmp = Float64(Float64(2.0 / Float64(t_m / l)) / Float64(Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * k), k, Float64(Float64(2.0 * t_m) * t_m)) / l) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * t_m) / l) * k) * Float64(Float64(fma(Float64(Float64(k * k) * t_m), Float64(0.6666666666666666 + (Float64(t_m * t_m) ^ -1.0)), Float64(2.0 * t_m)) / l) * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.8e-78], N[(N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(0.6666666666666666 + N[Power[N[(t$95$m * t$95$m), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.8000000000000001e-78

   1. Initial program 48.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 72.0

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 72.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\sin k \cdot t\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

   6. Applied rewrites 77.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)\right)}} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{t}{\ell}}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right) \cdot \left(t \cdot \sin k\right)}} \]

      9. lower-*.f64 79.1

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right) \cdot \left(t \cdot \sin k\right)}} \]

   8. Applied rewrites 79.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(\frac{\tan k \cdot t}{\ell} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(\sin k \cdot t\right)}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}}} \]

   11. Applied rewrites 65.2%

       \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot k, k, \left(2 \cdot t\right) \cdot t\right)}{\ell} \cdot \left(k \cdot k\right)}} \]

   if 1.8000000000000001e-78 < t

   1. Initial program 68.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 87.8

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 87.8%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\sin k \cdot t\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

   6. Applied rewrites 92.7%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \left(2 \cdot \frac{t}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(\left(2 \cdot \frac{t}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot k\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(\left(2 \cdot \frac{t}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot k\right)}} \]

   9. Applied rewrites 82.4%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, 0.6666666666666666 + \frac{1}{t \cdot t}, 2 \cdot t\right)}{\ell} \cdot k\right)}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{2}}{\ell}} \cdot \left(\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, \frac{2}{3} + \frac{1}{t \cdot t}, 2 \cdot t\right)}{\ell} \cdot k\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{2} \cdot k}}{\ell} \cdot \left(\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, \frac{2}{3} + \frac{1}{t \cdot t}, 2 \cdot t\right)}{\ell} \cdot k\right)} \]

       2. associate-*l/ N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot k\right)} \cdot \left(\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, \frac{2}{3} + \frac{1}{t \cdot t}, 2 \cdot t\right)}{\ell} \cdot k\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot k\right)} \cdot \left(\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, \frac{2}{3} + \frac{1}{t \cdot t}, 2 \cdot t\right)}{\ell} \cdot k\right)} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{2}}{\ell}} \cdot k\right) \cdot \left(\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, \frac{2}{3} + \frac{1}{t \cdot t}, 2 \cdot t\right)}{\ell} \cdot k\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot k\right) \cdot \left(\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, \frac{2}{3} + \frac{1}{t \cdot t}, 2 \cdot t\right)}{\ell} \cdot k\right)} \]

       6. lower-*.f64 76.2

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot k\right) \cdot \left(\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, 0.6666666666666666 + \frac{1}{t \cdot t}, 2 \cdot t\right)}{\ell} \cdot k\right)} \]

   12. Applied rewrites 76.2%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot k\right)} \cdot \left(\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, 0.6666666666666666 + \frac{1}{t \cdot t}, 2 \cdot t\right)}{\ell} \cdot k\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{2}{\frac{t}{\ell}}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot k, k, \left(2 \cdot t\right) \cdot t\right)}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot t}{\ell} \cdot k\right) \cdot \left(\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, 0.6666666666666666 + {\left(t \cdot t\right)}^{-1}, 2 \cdot t\right)}{\ell} \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 72.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-64}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \sin k\right)\right) \cdot \left(\frac{t\_m}{\ell} \cdot \left(2 \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell}}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot k, k, \left(2 \cdot t\_m\right) \cdot t\_m\right)}{\ell} \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2e-64)
    (/ 2.0 (* (* (/ t_m l) (* t_m (sin k))) (* (/ t_m l) (* 2.0 k))))
    (/
     (/ 2.0 (/ t_m l))
     (*
      (/
       (fma
        (* (fma 0.3333333333333333 (* t_m t_m) 1.0) k)
        k
        (* (* 2.0 t_m) t_m))
       l)
      (* k k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2e-64) {
		tmp = 2.0 / (((t_m / l) * (t_m * sin(k))) * ((t_m / l) * (2.0 * k)));
	} else {
		tmp = (2.0 / (t_m / l)) / ((fma((fma(0.3333333333333333, (t_m * t_m), 1.0) * k), k, ((2.0 * t_m) * t_m)) / l) * (k * k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2e-64)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(t_m * sin(k))) * Float64(Float64(t_m / l) * Float64(2.0 * k))));
	else
		tmp = Float64(Float64(2.0 / Float64(t_m / l)) / Float64(Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * k), k, Float64(Float64(2.0 * t_m) * t_m)) / l) * Float64(k * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2e-64], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.99999999999999993e-64

   1. Initial program 56.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 80.7

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 80.7%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\sin k \cdot t\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

   6. Applied rewrites 86.9%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(2 \cdot k\right)}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 76.6

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(2 \cdot k\right)}\right)} \]

   9. Applied rewrites 76.6%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(2 \cdot k\right)}\right)} \]

   if 1.99999999999999993e-64 < k

   1. Initial program 48.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 68.1

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 68.1%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\sin k \cdot t\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

   6. Applied rewrites 70.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)\right)}} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{t}{\ell}}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right) \cdot \left(t \cdot \sin k\right)}} \]

      9. lower-*.f64 75.1

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right) \cdot \left(t \cdot \sin k\right)}} \]

   8. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(\frac{\tan k \cdot t}{\ell} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(\sin k \cdot t\right)}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}}} \]

   11. Applied rewrites 66.7%

       \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot k, k, \left(2 \cdot t\right) \cdot t\right)}{\ell} \cdot \left(k \cdot k\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 74.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.5 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell}}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot k, k, \left(2 \cdot t\_m\right) \cdot t\_m\right)}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;t\_m \leq 1.1 \cdot 10^{+115}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t\_m}}{t\_m \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{t\_m}\right)}^{2}}{\left(t\_m \cdot k\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.5e-75)
    (/
     (/ 2.0 (/ t_m l))
     (*
      (/
       (fma
        (* (fma 0.3333333333333333 (* t_m t_m) 1.0) k)
        k
        (* (* 2.0 t_m) t_m))
       l)
      (* k k)))
    (if (<= t_m 1.1e+115)
      (/ (* l (/ (/ (/ l k) k) t_m)) (* t_m t_m))
      (/ (pow (/ l t_m) 2.0) (* (* t_m k) k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6.5e-75) {
		tmp = (2.0 / (t_m / l)) / ((fma((fma(0.3333333333333333, (t_m * t_m), 1.0) * k), k, ((2.0 * t_m) * t_m)) / l) * (k * k));
	} else if (t_m <= 1.1e+115) {
		tmp = (l * (((l / k) / k) / t_m)) / (t_m * t_m);
	} else {
		tmp = pow((l / t_m), 2.0) / ((t_m * k) * k);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 6.5e-75)
		tmp = Float64(Float64(2.0 / Float64(t_m / l)) / Float64(Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * k), k, Float64(Float64(2.0 * t_m) * t_m)) / l) * Float64(k * k)));
	elseif (t_m <= 1.1e+115)
		tmp = Float64(Float64(l * Float64(Float64(Float64(l / k) / k) / t_m)) / Float64(t_m * t_m));
	else
		tmp = Float64((Float64(l / t_m) ^ 2.0) / Float64(Float64(t_m * k) * k));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 6.5e-75], N[(N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.1e+115], N[(N[(l * N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(l / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(t$95$m * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 6.5000000000000002e-75

   1. Initial program 48.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 72.0

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 72.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\sin k \cdot t\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

   6. Applied rewrites 77.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)\right)}} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{t}{\ell}}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right) \cdot \left(t \cdot \sin k\right)}} \]

      9. lower-*.f64 79.1

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right) \cdot \left(t \cdot \sin k\right)}} \]

   8. Applied rewrites 79.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(\frac{\tan k \cdot t}{\ell} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(\sin k \cdot t\right)}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}}} \]

   11. Applied rewrites 65.2%

       \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot k, k, \left(2 \cdot t\right) \cdot t\right)}{\ell} \cdot \left(k \cdot k\right)}} \]

   if 6.5000000000000002e-75 < t < 1.1e115

   1. Initial program 80.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 68.7

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 68.7%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 68.8%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 70.9%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 78.5%

       \[\leadsto \frac{\ell \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}}{\color{blue}{t \cdot t}} \]

   if 1.1e115 < t

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 46.4

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 46.4%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 46.4%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.1%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 85.5%

       \[\leadsto \frac{-{\left(\frac{\ell}{t}\right)}^{2}}{\color{blue}{\left(\left(-t\right) \cdot k\right) \cdot k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.5 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{2}{\frac{t}{\ell}}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot k, k, \left(2 \cdot t\right) \cdot t\right)}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+115}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{t}\right)}^{2}}{\left(t \cdot k\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 74.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.5 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell}}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot k, k, \left(2 \cdot t\_m\right) \cdot t\_m\right)}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;t\_m \leq 7.5 \cdot 10^{+118}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t\_m}}{t\_m \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \ell}{{\left(k \cdot t\_m\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.5e-75)
    (/
     (/ 2.0 (/ t_m l))
     (*
      (/
       (fma
        (* (fma 0.3333333333333333 (* t_m t_m) 1.0) k)
        k
        (* (* 2.0 t_m) t_m))
       l)
      (* k k)))
    (if (<= t_m 7.5e+118)
      (/ (* l (/ (/ (/ l k) k) t_m)) (* t_m t_m))
      (/ (* (/ l t_m) l) (pow (* k t_m) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6.5e-75) {
		tmp = (2.0 / (t_m / l)) / ((fma((fma(0.3333333333333333, (t_m * t_m), 1.0) * k), k, ((2.0 * t_m) * t_m)) / l) * (k * k));
	} else if (t_m <= 7.5e+118) {
		tmp = (l * (((l / k) / k) / t_m)) / (t_m * t_m);
	} else {
		tmp = ((l / t_m) * l) / pow((k * t_m), 2.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 6.5e-75)
		tmp = Float64(Float64(2.0 / Float64(t_m / l)) / Float64(Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * k), k, Float64(Float64(2.0 * t_m) * t_m)) / l) * Float64(k * k)));
	elseif (t_m <= 7.5e+118)
		tmp = Float64(Float64(l * Float64(Float64(Float64(l / k) / k) / t_m)) / Float64(t_m * t_m));
	else
		tmp = Float64(Float64(Float64(l / t_m) * l) / (Float64(k * t_m) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 6.5e-75], N[(N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.5e+118], N[(N[(l * N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 6.5000000000000002e-75

   1. Initial program 48.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 72.0

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 72.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\sin k \cdot t\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

   6. Applied rewrites 77.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)\right)}} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{t}{\ell}}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right) \cdot \left(t \cdot \sin k\right)}} \]

      9. lower-*.f64 79.1

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right) \cdot \left(t \cdot \sin k\right)}} \]

   8. Applied rewrites 79.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(\frac{\tan k \cdot t}{\ell} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(\sin k \cdot t\right)}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}}} \]

   11. Applied rewrites 65.2%

       \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot k, k, \left(2 \cdot t\right) \cdot t\right)}{\ell} \cdot \left(k \cdot k\right)}} \]

   if 6.5000000000000002e-75 < t < 7.50000000000000003e118

   1. Initial program 80.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 68.7

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 68.7%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 68.8%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 70.9%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 78.5%

       \[\leadsto \frac{\ell \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}}{\color{blue}{t \cdot t}} \]

   if 7.50000000000000003e118 < t

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 46.4

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 46.4%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 46.4%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.1%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 82.6%

       \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{{\left(k \cdot t\right)}^{2}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 67.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.15 \cdot 10^{-177}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k} \cdot \frac{\ell}{k \cdot t\_m}\\ \mathbf{elif}\;k \leq 1.95 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m}}{t\_m \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell}}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot k, k, \left(2 \cdot t\_m\right) \cdot t\_m\right)}{\ell} \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.15e-177)
    (* (/ (/ l (* t_m t_m)) k) (/ l (* k t_m)))
    (if (<= k 1.95e-64)
      (/ (* (/ l t_m) (/ l t_m)) (* t_m (* k k)))
      (/
       (/ 2.0 (/ t_m l))
       (*
        (/
         (fma
          (* (fma 0.3333333333333333 (* t_m t_m) 1.0) k)
          k
          (* (* 2.0 t_m) t_m))
         l)
        (* k k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.15e-177) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	} else if (k <= 1.95e-64) {
		tmp = ((l / t_m) * (l / t_m)) / (t_m * (k * k));
	} else {
		tmp = (2.0 / (t_m / l)) / ((fma((fma(0.3333333333333333, (t_m * t_m), 1.0) * k), k, ((2.0 * t_m) * t_m)) / l) * (k * k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.15e-177)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / k) * Float64(l / Float64(k * t_m)));
	elseif (k <= 1.95e-64)
		tmp = Float64(Float64(Float64(l / t_m) * Float64(l / t_m)) / Float64(t_m * Float64(k * k)));
	else
		tmp = Float64(Float64(2.0 / Float64(t_m / l)) / Float64(Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * k), k, Float64(Float64(2.0 * t_m) * t_m)) / l) * Float64(k * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.15e-177], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.95e-64], N[(N[(N[(l / t$95$m), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.1500000000000001e-177

   1. Initial program 55.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 53.9

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 53.9%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 53.9%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 55.3%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 66.0%

       \[\leadsto \frac{\frac{\ell}{t \cdot t}}{k} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

   if 2.1500000000000001e-177 < k < 1.9499999999999998e-64

   1. Initial program 67.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 73.2

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 73.2%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.3%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 87.2%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.9%

       \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{t} \cdot \left(k \cdot k\right)} \]

   if 1.9499999999999998e-64 < k

   1. Initial program 48.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 68.1

          \[\leadsto \frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 68.1%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot t\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\sin k \cdot t\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \sin k\right)}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

   6. Applied rewrites 70.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right)} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)\right)}} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{t}{\ell}}}}{\left(t \cdot \sin k\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right) \cdot \left(t \cdot \sin k\right)}} \]

      9. lower-*.f64 75.1

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right) \cdot \left(t \cdot \sin k\right)}} \]

   8. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\left(\frac{\tan k \cdot t}{\ell} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(\sin k \cdot t\right)}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot {k}^{2}}} \]

   11. Applied rewrites 66.7%

       \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot k, k, \left(2 \cdot t\right) \cdot t\right)}{\ell} \cdot \left(k \cdot k\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 69.5% accurate, 6.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-179}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t\_m}}{t\_m}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m} \cdot \frac{\ell}{k}}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m}}{t\_m \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 2e-179)
    (/ (* (/ l t_m) (/ (/ (/ l k) k) t_m)) t_m)
    (if (<= (* l l) 5e+74)
      (/ (* (/ l (* t_m t_m)) (/ l k)) (* t_m k))
      (/ (* (/ l t_m) (/ l t_m)) (* t_m (* k k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 2e-179) {
		tmp = ((l / t_m) * (((l / k) / k) / t_m)) / t_m;
	} else if ((l * l) <= 5e+74) {
		tmp = ((l / (t_m * t_m)) * (l / k)) / (t_m * k);
	} else {
		tmp = ((l / t_m) * (l / t_m)) / (t_m * (k * k));
	}
	return t_s * tmp;
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 2d-179) then
        tmp = ((l / t_m) * (((l / k) / k) / t_m)) / t_m
    else if ((l * l) <= 5d+74) then
        tmp = ((l / (t_m * t_m)) * (l / k)) / (t_m * k)
    else
        tmp = ((l / t_m) * (l / t_m)) / (t_m * (k * k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 2e-179) {
		tmp = ((l / t_m) * (((l / k) / k) / t_m)) / t_m;
	} else if ((l * l) <= 5e+74) {
		tmp = ((l / (t_m * t_m)) * (l / k)) / (t_m * k);
	} else {
		tmp = ((l / t_m) * (l / t_m)) / (t_m * (k * k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 2e-179:
		tmp = ((l / t_m) * (((l / k) / k) / t_m)) / t_m
	elif (l * l) <= 5e+74:
		tmp = ((l / (t_m * t_m)) * (l / k)) / (t_m * k)
	else:
		tmp = ((l / t_m) * (l / t_m)) / (t_m * (k * k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 2e-179)
		tmp = Float64(Float64(Float64(l / t_m) * Float64(Float64(Float64(l / k) / k) / t_m)) / t_m);
	elseif (Float64(l * l) <= 5e+74)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) * Float64(l / k)) / Float64(t_m * k));
	else
		tmp = Float64(Float64(Float64(l / t_m) * Float64(l / t_m)) / Float64(t_m * Float64(k * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 2e-179)
		tmp = ((l / t_m) * (((l / k) / k) / t_m)) / t_m;
	elseif ((l * l) <= 5e+74)
		tmp = ((l / (t_m * t_m)) * (l / k)) / (t_m * k);
	else
		tmp = ((l / t_m) * (l / t_m)) / (t_m * (k * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 2e-179], N[(N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 5e+74], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 2e-179

   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. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 58.5

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 58.5%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 58.5%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.5%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 82.4%

       \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}}{\color{blue}{t}} \]

   if 2e-179 < (*.f64 l l) < 4.99999999999999963e74

   1. Initial program 71.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 60.2

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 60.2%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 73.5%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \frac{\ell}{k}}{\color{blue}{t \cdot k}} \]

   if 4.99999999999999963e74 < (*.f64 l l)

   1. Initial program 44.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 47.9

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 47.9%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 47.9%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 52.5%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 60.1%

       \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{t} \cdot \left(k \cdot k\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 65.1% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k} \cdot \frac{\ell}{k \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\ell}{k} \cdot \ell}{t\_m \cdot \left(\left(k \cdot t\_m\right) \cdot t\_m\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6.5e+90)
    (* (/ (/ l (* t_m t_m)) k) (/ l (* k t_m)))
    (/ (* (/ (- l) k) l) (* t_m (* (* k t_m) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.5e+90) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	} else {
		tmp = ((-l / k) * l) / (t_m * ((k * t_m) * t_m));
	}
	return t_s * tmp;
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6.5d+90) then
        tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m))
    else
        tmp = ((-l / k) * l) / (t_m * ((k * t_m) * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.5e+90) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	} else {
		tmp = ((-l / k) * l) / (t_m * ((k * t_m) * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 6.5e+90:
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m))
	else:
		tmp = ((-l / k) * l) / (t_m * ((k * t_m) * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6.5e+90)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / k) * Float64(l / Float64(k * t_m)));
	else
		tmp = Float64(Float64(Float64(Float64(-l) / k) * l) / Float64(t_m * Float64(Float64(k * t_m) * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 6.5e+90)
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	else
		tmp = ((-l / k) * l) / (t_m * ((k * t_m) * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6.5e+90], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-l) / k), $MachinePrecision] * l), $MachinePrecision] / N[(t$95$m * N[(N[(k * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.5000000000000001e90

   1. Initial program 56.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 56.9

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 56.9%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 59.3%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 66.7%

       \[\leadsto \frac{\frac{\ell}{t \cdot t}}{k} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

   if 6.5000000000000001e90 < k

   1. Initial program 39.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 41.0

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 41.0%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 45.5%

      \[\leadsto \frac{\frac{\ell}{k} \cdot \left(-\ell\right)}{\color{blue}{k \cdot {t}^{3}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 47.9%

      \[\leadsto \frac{\frac{\ell}{k} \cdot \left(-\ell\right)}{t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot t}}{k} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\ell}{k} \cdot \ell}{t \cdot \left(\left(k \cdot t\right) \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 61.4% accurate, 9.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m} \cdot \ell}{\left(k \cdot t\_m\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\ell}{k} \cdot \ell}{t\_m \cdot \left(\left(k \cdot t\_m\right) \cdot t\_m\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6.5e+90)
    (/ (* (/ l (* t_m t_m)) l) (* (* k t_m) k))
    (/ (* (/ (- l) k) l) (* t_m (* (* k t_m) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.5e+90) {
		tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k);
	} else {
		tmp = ((-l / k) * l) / (t_m * ((k * t_m) * t_m));
	}
	return t_s * tmp;
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6.5d+90) then
        tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k)
    else
        tmp = ((-l / k) * l) / (t_m * ((k * t_m) * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.5e+90) {
		tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k);
	} else {
		tmp = ((-l / k) * l) / (t_m * ((k * t_m) * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 6.5e+90:
		tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k)
	else:
		tmp = ((-l / k) * l) / (t_m * ((k * t_m) * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6.5e+90)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) * l) / Float64(Float64(k * t_m) * k));
	else
		tmp = Float64(Float64(Float64(Float64(-l) / k) * l) / Float64(t_m * Float64(Float64(k * t_m) * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 6.5e+90)
		tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k);
	else
		tmp = ((-l / k) * l) / (t_m * ((k * t_m) * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6.5e+90], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-l) / k), $MachinePrecision] * l), $MachinePrecision] / N[(t$95$m * N[(N[(k * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.5000000000000001e90

   1. Initial program 56.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 56.9

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 56.9%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 59.3%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 63.9%

       \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\left(k \cdot t\right) \cdot \color{blue}{k}} \]

   if 6.5000000000000001e90 < k

   1. Initial program 39.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 41.0

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 41.0%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 45.5%

      \[\leadsto \frac{\frac{\ell}{k} \cdot \left(-\ell\right)}{\color{blue}{k \cdot {t}^{3}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 47.9%

      \[\leadsto \frac{\frac{\ell}{k} \cdot \left(-\ell\right)}{t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot t} \cdot \ell}{\left(k \cdot t\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\ell}{k} \cdot \ell}{t \cdot \left(\left(k \cdot t\right) \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 60.6% accurate, 9.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m} \cdot \ell}{\left(k \cdot t\_m\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\ell}{\left(t\_m \cdot t\_m\right) \cdot t\_m} \cdot \frac{\ell}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6.6e+90)
    (/ (* (/ l (* t_m t_m)) l) (* (* k t_m) k))
    (* (/ (- l) (* (* t_m t_m) t_m)) (/ l (* k k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.6e+90) {
		tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k);
	} else {
		tmp = (-l / ((t_m * t_m) * t_m)) * (l / (k * k));
	}
	return t_s * tmp;
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6.6d+90) then
        tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k)
    else
        tmp = (-l / ((t_m * t_m) * t_m)) * (l / (k * k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.6e+90) {
		tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k);
	} else {
		tmp = (-l / ((t_m * t_m) * t_m)) * (l / (k * k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 6.6e+90:
		tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k)
	else:
		tmp = (-l / ((t_m * t_m) * t_m)) * (l / (k * k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6.6e+90)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) * l) / Float64(Float64(k * t_m) * k));
	else
		tmp = Float64(Float64(Float64(-l) / Float64(Float64(t_m * t_m) * t_m)) * Float64(l / Float64(k * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 6.6e+90)
		tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k);
	else
		tmp = (-l / ((t_m * t_m) * t_m)) * (l / (k * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6.6e+90], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(N[((-l) / N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.60000000000000016e90

   1. Initial program 56.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 56.9

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 56.9%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 59.3%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 63.9%

       \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\left(k \cdot t\right) \cdot \color{blue}{k}} \]

   if 6.60000000000000016e90 < k

   1. Initial program 39.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 41.0

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 41.0%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 43.0%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(-t\right)} \cdot \frac{\ell}{k \cdot k} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot t} \cdot \ell}{\left(k \cdot t\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 61.2% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\frac{\ell}{t\_m \cdot t\_m} \cdot \ell}{\left(k \cdot t\_m\right) \cdot k} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (* (/ l (* t_m t_m)) l) (* (* k t_m) k))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (((l / (t_m * t_m)) * l) / ((k * t_m) * k));
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (((l / (t_m * t_m)) * l) / ((k * t_m) * k))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (((l / (t_m * t_m)) * l) / ((k * t_m) * k));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (((l / (t_m * t_m)) * l) / ((k * t_m) * k))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(Float64(l / Float64(t_m * t_m)) * l) / Float64(Float64(k * t_m) * k)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (((l / (t_m * t_m)) * l) / ((k * t_m) * k));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   9. lower-*.f64 54.3

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

5. Applied rewrites 54.3%

   \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
6. Step-by-step derivation

7. Applied rewrites 54.3%

   \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
8. Step-by-step derivation

9. Applied rewrites 56.0%

   \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]
10. Step-by-step derivation

11. Applied rewrites 59.5%

    \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\left(k \cdot t\right) \cdot \color{blue}{k}} \]
12. Add Preprocessing

Alternative 24: 58.9% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\frac{\ell}{t\_m \cdot t\_m}}{\left(k \cdot k\right) \cdot t\_m}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* l (/ (/ l (* t_m t_m)) (* (* k k) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * ((l / (t_m * t_m)) / ((k * k) * t_m)));
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (l * ((l / (t_m * t_m)) / ((k * k) * t_m)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * ((l / (t_m * t_m)) / ((k * k) * t_m)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * ((l / (t_m * t_m)) / ((k * k) * t_m)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(Float64(l / Float64(t_m * t_m)) / Float64(Float64(k * k) * t_m))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (l * ((l / (t_m * t_m)) / ((k * k) * t_m)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(l * N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   9. lower-*.f64 54.3

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

5. Applied rewrites 54.3%

   \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
6. Step-by-step derivation

7. Applied rewrites 54.3%

   \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
8. Step-by-step derivation

9. Applied rewrites 56.0%

   \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]
10. Step-by-step derivation

11. Applied rewrites 56.1%

    \[\leadsto \ell \cdot \color{blue}{\frac{\frac{\ell}{t \cdot t}}{\left(k \cdot k\right) \cdot t}} \]
12. Add Preprocessing

Alternative 25: 54.5% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\ell \cdot \ell}{\left(t\_m \cdot t\_m\right) \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (* l l) (* (* t_m t_m) (* (* k k) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) / ((t_m * t_m) * ((k * k) * t_m)));
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l * l) / ((t_m * t_m) * ((k * k) * t_m)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) / ((t_m * t_m) * ((k * k) * t_m)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l * l) / ((t_m * t_m) * ((k * k) * t_m)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l * l) / Float64(Float64(t_m * t_m) * Float64(Float64(k * k) * t_m))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l * l) / ((t_m * t_m) * ((k * k) * t_m)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   9. lower-*.f64 54.3

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

5. Applied rewrites 54.3%

   \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
6. Step-by-step derivation

7. Applied rewrites 54.3%

   \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
8. Step-by-step derivation

9. Applied rewrites 56.0%

   \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]
10. Step-by-step derivation

11. Applied rewrites 49.9%

    \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024358 
(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))))