Toniolo and Linder, Equation (10+)

Percentage Accurate: 53.8% → 86.7%

Time: 8.4s

Alternatives: 16

Speedup: 6.6×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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: 53.8% 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: 86.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{-38}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(\tan k \cdot \left(\frac{\sin k \cdot t\_m}{\ell} \cdot t\_m\right)\right)\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m} \cdot k, \frac{1}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

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 (<= t_m 1.8e-38)
    (* (/ 2.0 (/ (* (pow k 2.0) (* t_m (pow (sin k) 2.0))) (* l (cos k)))) l)
    (/
     2.0
     (*
      (* (/ t_m l) (* (tan k) (* (/ (* (sin k) t_m) l) t_m)))
      (fma (* (/ k t_m) k) (/ 1.0 t_m) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.8e-38) {
		tmp = (2.0 / ((pow(k, 2.0) * (t_m * pow(sin(k), 2.0))) / (l * cos(k)))) * l;
	} else {
		tmp = 2.0 / (((t_m / l) * (tan(k) * (((sin(k) * t_m) / l) * t_m))) * fma(((k / t_m) * k), (1.0 / t_m), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.8e-38)
		tmp = Float64(Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0))) / Float64(l * cos(k)))) * l);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(tan(k) * Float64(Float64(Float64(sin(k) * t_m) / l) * t_m))) * fma(Float64(Float64(k / t_m) * k), Float64(1.0 / t_m), 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.8e-38], N[(N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k / t$95$m), $MachinePrecision] * k), $MachinePrecision] * N[(1.0 / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.8e-38

   1. Initial program 53.8%

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

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.2

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

   3. Applied rewrites 66.2%

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

   5. Applied rewrites 53.0%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 65.2

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

   8. Applied rewrites 65.2%

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

   if 1.8e-38 < t

   1. Initial program 53.8%

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

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.2

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

   3. Applied rewrites 66.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. mult-flip N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 74.3

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

   7. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 76.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 76.0

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

   9. Applied rewrites 76.0%

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

Alternative 2: 86.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{-38}:\\ \;\;\;\;\left(2 \cdot \frac{\ell \cdot \cos k}{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(\tan k \cdot \left(\frac{\sin k \cdot t\_m}{\ell} \cdot t\_m\right)\right)\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m} \cdot k, \frac{1}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

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 (<= t_m 1.8e-38)
    (* (* 2.0 (/ (* l (cos k)) (* (pow k 2.0) (* t_m (pow (sin k) 2.0))))) l)
    (/
     2.0
     (*
      (* (/ t_m l) (* (tan k) (* (/ (* (sin k) t_m) l) t_m)))
      (fma (* (/ k t_m) k) (/ 1.0 t_m) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.8e-38) {
		tmp = (2.0 * ((l * cos(k)) / (pow(k, 2.0) * (t_m * pow(sin(k), 2.0))))) * l;
	} else {
		tmp = 2.0 / (((t_m / l) * (tan(k) * (((sin(k) * t_m) / l) * t_m))) * fma(((k / t_m) * k), (1.0 / t_m), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.8e-38)
		tmp = Float64(Float64(2.0 * Float64(Float64(l * cos(k)) / Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0))))) * l);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(tan(k) * Float64(Float64(Float64(sin(k) * t_m) / l) * t_m))) * fma(Float64(Float64(k / t_m) * k), Float64(1.0 / t_m), 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.8e-38], N[(N[(2.0 * N[(N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k / t$95$m), $MachinePrecision] * k), $MachinePrecision] * N[(1.0 / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.8e-38

   1. Initial program 53.8%

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

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.2

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

   3. Applied rewrites 66.2%

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

   5. Applied rewrites 53.0%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 65.2

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

   8. Applied rewrites 65.2%

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

   if 1.8e-38 < t

   1. Initial program 53.8%

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

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.2

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

   3. Applied rewrites 66.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. mult-flip N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 74.3

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

   7. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 76.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 76.0

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

   9. Applied rewrites 76.0%

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

Alternative 3: 79.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k \cdot t\_m}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.5 \cdot 10^{-244}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot t\_2\right)\right) \cdot \tan k\right) \cdot 2}\\ \mathbf{elif}\;t\_m \leq 1.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \frac{k \cdot t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\left(t\_2 \cdot t\_m\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right) \cdot \tan k\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (* (sin k) t_m) l)))
   (*
    t_s
    (if (<= t_m 6.5e-244)
      (/ 2.0 (* (* (* (/ t_m l) (* t_m t_2)) (tan k)) 2.0))
      (if (<= t_m 1.5e-96)
        (/
         2.0
         (*
          (* (* (/ t_m l) (* t_m (/ (* k t_m) l))) (tan k))
          (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
        (/
         2.0
         (*
          (/ t_m l)
          (* (* t_2 t_m) (* (fma (/ k (* t_m t_m)) k 2.0) (tan k))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (sin(k) * t_m) / l;
	double tmp;
	if (t_m <= 6.5e-244) {
		tmp = 2.0 / ((((t_m / l) * (t_m * t_2)) * tan(k)) * 2.0);
	} else if (t_m <= 1.5e-96) {
		tmp = 2.0 / ((((t_m / l) * (t_m * ((k * t_m) / l))) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / ((t_m / l) * ((t_2 * t_m) * (fma((k / (t_m * t_m)), k, 2.0) * tan(k))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(sin(k) * t_m) / l)
	tmp = 0.0
	if (t_m <= 6.5e-244)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * Float64(t_m * t_2)) * tan(k)) * 2.0));
	elseif (t_m <= 1.5e-96)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * Float64(t_m * Float64(Float64(k * t_m) / l))) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(t_2 * t_m) * Float64(fma(Float64(k / Float64(t_m * t_m)), k, 2.0) * tan(k)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6.5e-244], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * t$95$2), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.5e-96], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $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], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$2 * t$95$m), $MachinePrecision] * N[(N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 6.4999999999999994e-244

   1. Initial program 53.8%

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

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.2

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

   3. Applied rewrites 66.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 67.5%

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

   if 6.4999999999999994e-244 < t < 1.5e-96

   1. Initial program 53.8%

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

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.2

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

   3. Applied rewrites 66.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 69.4%

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

   if 1.5e-96 < t

   1. Initial program 53.8%

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

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.2

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

   3. Applied rewrites 66.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. metadata-eval N/A

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

      14. lift-fma.f64 N/A

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

   7. Applied rewrites 69.6%

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

Alternative 4: 78.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k \cdot t\_m}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{-210}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot t\_2\right)\right) \cdot \tan k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(\tan k \cdot \left(t\_2 \cdot t\_m\right)\right)\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m} \cdot k, \frac{1}{t\_m}, 2\right)}\\ \end{array} \end{array} \end{array} \]

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) l)))
   (*
    t_s
    (if (<= t_m 2.9e-210)
      (/ 2.0 (* (* (* (/ t_m l) (* t_m t_2)) (tan k)) 2.0))
      (/
       2.0
       (*
        (* (/ t_m l) (* (tan k) (* t_2 t_m)))
        (fma (* (/ k t_m) k) (/ 1.0 t_m) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (sin(k) * t_m) / l;
	double tmp;
	if (t_m <= 2.9e-210) {
		tmp = 2.0 / ((((t_m / l) * (t_m * t_2)) * tan(k)) * 2.0);
	} else {
		tmp = 2.0 / (((t_m / l) * (tan(k) * (t_2 * t_m))) * fma(((k / t_m) * k), (1.0 / t_m), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(sin(k) * t_m) / l)
	tmp = 0.0
	if (t_m <= 2.9e-210)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * Float64(t_m * t_2)) * tan(k)) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(tan(k) * Float64(t_2 * t_m))) * fma(Float64(Float64(k / t_m) * k), Float64(1.0 / t_m), 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.9e-210], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * t$95$2), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$2 * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k / t$95$m), $MachinePrecision] * k), $MachinePrecision] * N[(1.0 / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.90000000000000006e-210

   1. Initial program 53.8%

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

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.2

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

   3. Applied rewrites 66.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 67.5%

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

   if 2.90000000000000006e-210 < t

   1. Initial program 53.8%

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

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.2

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

   3. Applied rewrites 66.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. mult-flip N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 74.3

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

   7. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 76.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 76.0

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

   9. Applied rewrites 76.0%

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

Alternative 5: 75.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sin k \cdot t\_m\\ t_3 := \frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \frac{t\_2}{\ell}\right)\right) \cdot \tan k\right) \cdot 2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.5 \cdot 10^{-244}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_m \leq 2.65 \cdot 10^{-89}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \frac{k \cdot t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;t\_m \leq 3.9 \cdot 10^{+212}:\\ \;\;\;\;\frac{\frac{\ell + \ell}{\left(\left(t\_2 \cdot t\_m\right) \cdot \tan k\right) \cdot \frac{t\_m}{\ell}}}{\mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (sin k) t_m))
        (t_3 (/ 2.0 (* (* (* (/ t_m l) (* t_m (/ t_2 l))) (tan k)) 2.0))))
   (*
    t_s
    (if (<= t_m 6.5e-244)
      t_3
      (if (<= t_m 2.65e-89)
        (/
         2.0
         (*
          (* (* (/ t_m l) (* t_m (/ (* k t_m) l))) (tan k))
          (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
        (if (<= t_m 3.9e+212)
          (/
           (/ (+ l l) (* (* (* t_2 t_m) (tan k)) (/ t_m l)))
           (fma (/ k (* t_m t_m)) k 2.0))
          t_3))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = sin(k) * t_m;
	double t_3 = 2.0 / ((((t_m / l) * (t_m * (t_2 / l))) * tan(k)) * 2.0);
	double tmp;
	if (t_m <= 6.5e-244) {
		tmp = t_3;
	} else if (t_m <= 2.65e-89) {
		tmp = 2.0 / ((((t_m / l) * (t_m * ((k * t_m) / l))) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else if (t_m <= 3.9e+212) {
		tmp = ((l + l) / (((t_2 * t_m) * tan(k)) * (t_m / l))) / fma((k / (t_m * t_m)), k, 2.0);
	} else {
		tmp = t_3;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(sin(k) * t_m)
	t_3 = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * Float64(t_m * Float64(t_2 / l))) * tan(k)) * 2.0))
	tmp = 0.0
	if (t_m <= 6.5e-244)
		tmp = t_3;
	elseif (t_m <= 2.65e-89)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * Float64(t_m * Float64(Float64(k * t_m) / l))) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	elseif (t_m <= 3.9e+212)
		tmp = Float64(Float64(Float64(l + l) / Float64(Float64(Float64(t_2 * t_m) * tan(k)) * Float64(t_m / l))) / fma(Float64(k / Float64(t_m * t_m)), k, 2.0));
	else
		tmp = t_3;
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6.5e-244], t$95$3, If[LessEqual[t$95$m, 2.65e-89], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $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], If[LessEqual[t$95$m, 3.9e+212], N[(N[(N[(l + l), $MachinePrecision] / N[(N[(N[(t$95$2 * t$95$m), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision]), $MachinePrecision], t$95$3]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 6.4999999999999994e-244 or 3.9000000000000001e212 < t

   1. Initial program 53.8%

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

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.2

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

   3. Applied rewrites 66.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 67.5%

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

   if 6.4999999999999994e-244 < t < 2.65e-89

   1. Initial program 53.8%

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

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.2

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

   3. Applied rewrites 66.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 69.4%

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

   if 2.65e-89 < t < 3.9000000000000001e212

   1. Initial program 53.8%

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

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.2

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

   3. Applied rewrites 66.2%

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

   5. Applied rewrites 53.0%

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

   7. Applied rewrites 60.0%

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

Alternative 6: 75.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sin k \cdot t\_m\\ t_3 := \frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \frac{t\_2}{\ell}\right)\right) \cdot \tan k\right) \cdot 2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.5 \cdot 10^{-244}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_m \leq 3 \cdot 10^{-90}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \frac{k \cdot t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;t\_m \leq 3.9 \cdot 10^{+212}:\\ \;\;\;\;\frac{\ell + \ell}{\mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right) \cdot \left(\left(\left(t\_2 \cdot t\_m\right) \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (sin k) t_m))
        (t_3 (/ 2.0 (* (* (* (/ t_m l) (* t_m (/ t_2 l))) (tan k)) 2.0))))
   (*
    t_s
    (if (<= t_m 6.5e-244)
      t_3
      (if (<= t_m 3e-90)
        (/
         2.0
         (*
          (* (* (/ t_m l) (* t_m (/ (* k t_m) l))) (tan k))
          (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
        (if (<= t_m 3.9e+212)
          (/
           (+ l l)
           (*
            (fma (/ k (* t_m t_m)) k 2.0)
            (* (* (* t_2 t_m) (tan k)) (/ t_m l))))
          t_3))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = sin(k) * t_m;
	double t_3 = 2.0 / ((((t_m / l) * (t_m * (t_2 / l))) * tan(k)) * 2.0);
	double tmp;
	if (t_m <= 6.5e-244) {
		tmp = t_3;
	} else if (t_m <= 3e-90) {
		tmp = 2.0 / ((((t_m / l) * (t_m * ((k * t_m) / l))) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else if (t_m <= 3.9e+212) {
		tmp = (l + l) / (fma((k / (t_m * t_m)), k, 2.0) * (((t_2 * t_m) * tan(k)) * (t_m / l)));
	} else {
		tmp = t_3;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(sin(k) * t_m)
	t_3 = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * Float64(t_m * Float64(t_2 / l))) * tan(k)) * 2.0))
	tmp = 0.0
	if (t_m <= 6.5e-244)
		tmp = t_3;
	elseif (t_m <= 3e-90)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * Float64(t_m * Float64(Float64(k * t_m) / l))) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	elseif (t_m <= 3.9e+212)
		tmp = Float64(Float64(l + l) / Float64(fma(Float64(k / Float64(t_m * t_m)), k, 2.0) * Float64(Float64(Float64(t_2 * t_m) * tan(k)) * Float64(t_m / l))));
	else
		tmp = t_3;
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6.5e-244], t$95$3, If[LessEqual[t$95$m, 3e-90], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $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], If[LessEqual[t$95$m, 3.9e+212], N[(N[(l + l), $MachinePrecision] / N[(N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[(N[(N[(t$95$2 * t$95$m), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 6.4999999999999994e-244 or 3.9000000000000001e212 < t

   1. Initial program 53.8%

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

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.2

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

   3. Applied rewrites 66.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 67.5%

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

   if 6.4999999999999994e-244 < t < 3.0000000000000002e-90

   1. Initial program 53.8%

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

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.2

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

   3. Applied rewrites 66.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 69.4%

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

   if 3.0000000000000002e-90 < t < 3.9000000000000001e212

   1. Initial program 53.8%

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

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.2

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

   3. Applied rewrites 66.2%

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

   5. Applied rewrites 53.0%

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

   7. Applied rewrites 60.3%

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

Alternative 7: 72.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 7.8 \cdot 10^{+56}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \frac{k \cdot t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m} \cdot k, \frac{1}{t\_m}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \frac{\sin k \cdot t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 7.8e+56)
    (/
     2.0
     (*
      (* (* (/ t_m l) (* t_m (/ (* k t_m) l))) (tan k))
      (fma (* (/ k t_m) k) (/ 1.0 t_m) 2.0)))
    (/ 2.0 (* (* (* (/ t_m l) (* t_m (/ (* (sin k) 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 tmp;
	if (l <= 7.8e+56) {
		tmp = 2.0 / ((((t_m / l) * (t_m * ((k * t_m) / l))) * tan(k)) * fma(((k / t_m) * k), (1.0 / t_m), 2.0));
	} else {
		tmp = 2.0 / ((((t_m / l) * (t_m * ((sin(k) * t_m) / l))) * 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)
	tmp = 0.0
	if (l <= 7.8e+56)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * Float64(t_m * Float64(Float64(k * t_m) / l))) * tan(k)) * fma(Float64(Float64(k / t_m) * k), Float64(1.0 / t_m), 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * Float64(t_m * Float64(Float64(sin(k) * t_m) / l))) * tan(k)) * 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[l, 7.8e+56], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k / t$95$m), $MachinePrecision] * k), $MachinePrecision] * N[(1.0 / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 7.79999999999999989e56

   1. Initial program 53.8%

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

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.2

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

   3. Applied rewrites 66.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. mult-flip N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 74.3

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

   7. Applied rewrites 74.3%

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

   8. Taylor expanded in k around 0

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

      1. lower-*.f64 69.4

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

   10. Applied rewrites 69.4%

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

   if 7.79999999999999989e56 < l

   1. Initial program 53.8%

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

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.2

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

   3. Applied rewrites 66.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 67.5%

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

Alternative 8: 70.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.8 \cdot 10^{-200}:\\ \;\;\;\;\frac{\frac{2}{\sin k}}{t\_m \cdot \left(\left(\frac{t\_m}{\ell \cdot \ell} \cdot t\_m\right) \cdot \left(k + k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \frac{k \cdot t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m} \cdot k, \frac{1}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

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 (<= t_m 9.8e-200)
    (/ (/ 2.0 (sin k)) (* t_m (* (* (/ t_m (* l l)) t_m) (+ k k))))
    (/
     2.0
     (*
      (* (* (/ t_m l) (* t_m (/ (* k t_m) l))) (tan k))
      (fma (* (/ k t_m) k) (/ 1.0 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 <= 9.8e-200) {
		tmp = (2.0 / sin(k)) / (t_m * (((t_m / (l * l)) * t_m) * (k + k)));
	} else {
		tmp = 2.0 / ((((t_m / l) * (t_m * ((k * t_m) / l))) * tan(k)) * fma(((k / t_m) * k), (1.0 / 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 <= 9.8e-200)
		tmp = Float64(Float64(2.0 / sin(k)) / Float64(t_m * Float64(Float64(Float64(t_m / Float64(l * l)) * t_m) * Float64(k + k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * Float64(t_m * Float64(Float64(k * t_m) / l))) * tan(k)) * fma(Float64(Float64(k / t_m) * k), Float64(1.0 / 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, 9.8e-200], N[(N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k / t$95$m), $MachinePrecision] * k), $MachinePrecision] * N[(1.0 / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.7999999999999999e-200

   1. Initial program 53.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 57.6%

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

   4. Taylor expanded in k around 0

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

      1. lower-*.f64 55.4

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

   6. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 55.4

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

   8. Applied rewrites 60.5%

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

   if 9.7999999999999999e-200 < t

   1. Initial program 53.8%

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

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.2

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

   3. Applied rewrites 66.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. mult-flip N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 74.3

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

   7. Applied rewrites 74.3%

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

   8. Taylor expanded in k around 0

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

      1. lower-*.f64 69.4

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

   10. Applied rewrites 69.4%

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

Alternative 9: 68.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{2}{\sin k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.7 \cdot 10^{-194}:\\ \;\;\;\;\frac{t\_2}{t\_m \cdot \left(\left(\frac{t\_m}{\ell \cdot \ell} \cdot t\_m\right) \cdot \left(k + k\right)\right)}\\ \mathbf{elif}\;t\_m \leq 1.68 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t\_m}}{t\_m \cdot t\_m}\\ \mathbf{elif}\;t\_m \leq 22:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{\ell} \cdot \left(\left(\tan k \cdot \frac{k}{\ell}\right) \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)}\\ \mathbf{elif}\;t\_m \leq 1.15 \cdot 10^{+168}:\\ \;\;\;\;\frac{\ell}{\left(t\_m \cdot \left(t\_m \cdot \left(k \cdot t\_m\right)\right)\right) \cdot k} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \frac{1}{\left(\frac{t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot t\_m\right)\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \end{array} \]

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 (sin k))))
   (*
    t_s
    (if (<= t_m 3.7e-194)
      (/ t_2 (* t_m (* (* (/ t_m (* l l)) t_m) (+ k k))))
      (if (<= t_m 1.68e-105)
        (/ (/ (* l (/ l (* k k))) t_m) (* t_m t_m))
        (if (<= t_m 22.0)
          (/
           2.0
           (*
            (/ (* (* t_m t_m) t_m) l)
            (* (* (tan k) (/ k l)) (fma k (/ k (* t_m t_m)) 2.0))))
          (if (<= t_m 1.15e+168)
            (* (/ l (* (* t_m (* t_m (* k t_m))) k)) l)
            (*
             t_2
             (/ 1.0 (* (* (/ t_m l) (* (/ t_m l) t_m)) (* 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 t_2 = 2.0 / sin(k);
	double tmp;
	if (t_m <= 3.7e-194) {
		tmp = t_2 / (t_m * (((t_m / (l * l)) * t_m) * (k + k)));
	} else if (t_m <= 1.68e-105) {
		tmp = ((l * (l / (k * k))) / t_m) / (t_m * t_m);
	} else if (t_m <= 22.0) {
		tmp = 2.0 / ((((t_m * t_m) * t_m) / l) * ((tan(k) * (k / l)) * fma(k, (k / (t_m * t_m)), 2.0)));
	} else if (t_m <= 1.15e+168) {
		tmp = (l / ((t_m * (t_m * (k * t_m))) * k)) * l;
	} else {
		tmp = t_2 * (1.0 / (((t_m / l) * ((t_m / l) * t_m)) * (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)
	t_2 = Float64(2.0 / sin(k))
	tmp = 0.0
	if (t_m <= 3.7e-194)
		tmp = Float64(t_2 / Float64(t_m * Float64(Float64(Float64(t_m / Float64(l * l)) * t_m) * Float64(k + k))));
	elseif (t_m <= 1.68e-105)
		tmp = Float64(Float64(Float64(l * Float64(l / Float64(k * k))) / t_m) / Float64(t_m * t_m));
	elseif (t_m <= 22.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / l) * Float64(Float64(tan(k) * Float64(k / l)) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0))));
	elseif (t_m <= 1.15e+168)
		tmp = Float64(Float64(l / Float64(Float64(t_m * Float64(t_m * Float64(k * t_m))) * k)) * l);
	else
		tmp = Float64(t_2 * Float64(1.0 / Float64(Float64(Float64(t_m / l) * Float64(Float64(t_m / l) * t_m)) * Float64(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_] := Block[{t$95$2 = N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.7e-194], N[(t$95$2 / N[(t$95$m * N[(N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.68e-105], N[(N[(N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 22.0], N[(2.0 / N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.15e+168], N[(N[(l / N[(N[(t$95$m * N[(t$95$m * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(t$95$2 * N[(1.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if t < 3.70000000000000008e-194

   1. Initial program 53.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 57.6%

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

   4. Taylor expanded in k around 0

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

      1. lower-*.f64 55.4

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

   6. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 55.4

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

   8. Applied rewrites 60.5%

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

   if 3.70000000000000008e-194 < t < 1.68000000000000003e-105

   1. Initial program 53.8%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 50.6

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

   4. Applied rewrites 50.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 58.0

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

      17. lift-pow.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 58.0

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

   6. Applied rewrites 58.0%

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

   if 1.68000000000000003e-105 < t < 22

   1. Initial program 53.8%

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

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.2

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

   3. Applied rewrites 66.2%

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 47.6%

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

   if 22 < t < 1.15e168

   1. Initial program 53.8%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 50.6

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

   4. Applied rewrites 50.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 59.2

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. unpow3 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 59.2

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

   6. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.2

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

   8. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 63.1

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

   10. Applied rewrites 63.1%

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

   if 1.15e168 < t

   1. Initial program 53.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 57.6%

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

   4. Taylor expanded in k around 0

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

      1. lower-*.f64 55.4

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

   6. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 62.1

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

   8. Applied rewrites 62.1%

      \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot t\right)\right)} \cdot \left(2 \cdot k\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 10: 65.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{-205}:\\ \;\;\;\;\frac{\ell}{\left(t\_m \cdot \left(t\_m \cdot \left(k \cdot t\_m\right)\right)\right) \cdot k} \cdot \ell\\ \mathbf{elif}\;k \leq 2.4 \cdot 10^{+40}:\\ \;\;\;\;\frac{2}{\sin k} \cdot \frac{1}{\left(\frac{t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot t\_m\right)\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{1}{t\_m \cdot \left(t\_m \cdot \left(\left(\mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right) \cdot \tan k\right) \cdot \frac{t\_m}{\ell \cdot \ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.6e-205)
    (* (/ l (* (* t_m (* t_m (* k t_m))) k)) l)
    (if (<= k 2.4e+40)
      (* (/ 2.0 (sin k)) (/ 1.0 (* (* (/ t_m l) (* (/ t_m l) t_m)) (* 2.0 k))))
      (*
       (/ 2.0 k)
       (/
        1.0
        (*
         t_m
         (*
          t_m
          (* (* (fma (/ k (* t_m t_m)) k 2.0) (tan k)) (/ t_m (* l l)))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.6e-205) {
		tmp = (l / ((t_m * (t_m * (k * t_m))) * k)) * l;
	} else if (k <= 2.4e+40) {
		tmp = (2.0 / sin(k)) * (1.0 / (((t_m / l) * ((t_m / l) * t_m)) * (2.0 * k)));
	} else {
		tmp = (2.0 / k) * (1.0 / (t_m * (t_m * ((fma((k / (t_m * t_m)), k, 2.0) * tan(k)) * (t_m / (l * l))))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.6e-205)
		tmp = Float64(Float64(l / Float64(Float64(t_m * Float64(t_m * Float64(k * t_m))) * k)) * l);
	elseif (k <= 2.4e+40)
		tmp = Float64(Float64(2.0 / sin(k)) * Float64(1.0 / Float64(Float64(Float64(t_m / l) * Float64(Float64(t_m / l) * t_m)) * Float64(2.0 * k))));
	else
		tmp = Float64(Float64(2.0 / k) * Float64(1.0 / Float64(t_m * Float64(t_m * Float64(Float64(fma(Float64(k / Float64(t_m * t_m)), k, 2.0) * tan(k)) * Float64(t_m / Float64(l * l)))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.6e-205], N[(N[(l / N[(N[(t$95$m * N[(t$95$m * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], If[LessEqual[k, 2.4e+40], N[(N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / k), $MachinePrecision] * N[(1.0 / N[(t$95$m * N[(t$95$m * N[(N[(N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.5999999999999998e-205

   1. Initial program 53.8%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 50.6

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

   4. Applied rewrites 50.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 59.2

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. unpow3 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 59.2

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

   6. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.2

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

   8. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 63.1

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

   10. Applied rewrites 63.1%

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

   if 2.5999999999999998e-205 < k < 2.4e40

   1. Initial program 53.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 57.6%

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

   4. Taylor expanded in k around 0

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

      1. lower-*.f64 55.4

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

   6. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 62.1

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

   8. Applied rewrites 62.1%

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

   if 2.4e40 < k

   1. Initial program 53.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\sin k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{2}{\sin k} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{2}{\sin k} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2}{\sin k}} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   3. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\frac{2}{\sin k} \cdot \frac{1}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \tan k\right)}} \]

   4. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{k}} \cdot \frac{1}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \tan k\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 54.3%

      \[\leadsto \frac{2}{\color{blue}{k}} \cdot \frac{1}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{k} \cdot \frac{1}{\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \tan k\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{k} \cdot \frac{1}{\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right)} \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \tan k\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{k} \cdot \frac{1}{\color{blue}{\left(t \cdot t\right) \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \tan k\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{k} \cdot \frac{1}{\color{blue}{\left(t \cdot t\right)} \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \tan k\right)\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{k} \cdot \frac{1}{\color{blue}{t \cdot \left(t \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \tan k\right)\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{k} \cdot \frac{1}{\color{blue}{t \cdot \left(t \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \tan k\right)\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{k} \cdot \frac{1}{t \cdot \color{blue}{\left(t \cdot \left(\frac{t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \tan k\right)\right)\right)}} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{k} \cdot \frac{1}{t \cdot \left(t \cdot \color{blue}{\left(\left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \tan k\right) \cdot \frac{t}{\ell \cdot \ell}\right)}\right)} \]

      9. lower-*.f64 61.2

         \[\leadsto \frac{2}{k} \cdot \frac{1}{t \cdot \left(t \cdot \color{blue}{\left(\left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \tan k\right) \cdot \frac{t}{\ell \cdot \ell}\right)}\right)} \]

   8. Applied rewrites 54.7%

      \[\leadsto \frac{2}{k} \cdot \frac{1}{\color{blue}{t \cdot \left(t \cdot \left(\left(\mathsf{fma}\left(\frac{k}{t \cdot t}, k, 2\right) \cdot \tan k\right) \cdot \frac{t}{\ell \cdot \ell}\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 65.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{2}{\sin k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.7 \cdot 10^{-194}:\\ \;\;\;\;\frac{t\_2}{t\_m \cdot \left(\left(\frac{t\_m}{\ell \cdot \ell} \cdot t\_m\right) \cdot \left(k + k\right)\right)}\\ \mathbf{elif}\;t\_m \leq 5.7 \cdot 10^{+61}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t\_m \cdot t\_m}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \frac{1}{\left(\frac{t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot t\_m\right)\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \end{array} \]

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 (sin k))))
   (*
    t_s
    (if (<= t_m 3.7e-194)
      (/ t_2 (* t_m (* (* (/ t_m (* l l)) t_m) (+ k k))))
      (if (<= t_m 5.7e+61)
        (/ (/ (* l (/ l (* k k))) (* t_m t_m)) t_m)
        (* t_2 (/ 1.0 (* (* (/ t_m l) (* (/ t_m l) t_m)) (* 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 t_2 = 2.0 / sin(k);
	double tmp;
	if (t_m <= 3.7e-194) {
		tmp = t_2 / (t_m * (((t_m / (l * l)) * t_m) * (k + k)));
	} else if (t_m <= 5.7e+61) {
		tmp = ((l * (l / (k * k))) / (t_m * t_m)) / t_m;
	} else {
		tmp = t_2 * (1.0 / (((t_m / l) * ((t_m / l) * t_m)) * (2.0 * 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) :: t_2
    real(8) :: tmp
    t_2 = 2.0d0 / sin(k)
    if (t_m <= 3.7d-194) then
        tmp = t_2 / (t_m * (((t_m / (l * l)) * t_m) * (k + k)))
    else if (t_m <= 5.7d+61) then
        tmp = ((l * (l / (k * k))) / (t_m * t_m)) / t_m
    else
        tmp = t_2 * (1.0d0 / (((t_m / l) * ((t_m / l) * t_m)) * (2.0d0 * 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 t_2 = 2.0 / Math.sin(k);
	double tmp;
	if (t_m <= 3.7e-194) {
		tmp = t_2 / (t_m * (((t_m / (l * l)) * t_m) * (k + k)));
	} else if (t_m <= 5.7e+61) {
		tmp = ((l * (l / (k * k))) / (t_m * t_m)) / t_m;
	} else {
		tmp = t_2 * (1.0 / (((t_m / l) * ((t_m / l) * t_m)) * (2.0 * 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):
	t_2 = 2.0 / math.sin(k)
	tmp = 0
	if t_m <= 3.7e-194:
		tmp = t_2 / (t_m * (((t_m / (l * l)) * t_m) * (k + k)))
	elif t_m <= 5.7e+61:
		tmp = ((l * (l / (k * k))) / (t_m * t_m)) / t_m
	else:
		tmp = t_2 * (1.0 / (((t_m / l) * ((t_m / l) * t_m)) * (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)
	t_2 = Float64(2.0 / sin(k))
	tmp = 0.0
	if (t_m <= 3.7e-194)
		tmp = Float64(t_2 / Float64(t_m * Float64(Float64(Float64(t_m / Float64(l * l)) * t_m) * Float64(k + k))));
	elseif (t_m <= 5.7e+61)
		tmp = Float64(Float64(Float64(l * Float64(l / Float64(k * k))) / Float64(t_m * t_m)) / t_m);
	else
		tmp = Float64(t_2 * Float64(1.0 / Float64(Float64(Float64(t_m / l) * Float64(Float64(t_m / l) * t_m)) * Float64(2.0 * 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)
	t_2 = 2.0 / sin(k);
	tmp = 0.0;
	if (t_m <= 3.7e-194)
		tmp = t_2 / (t_m * (((t_m / (l * l)) * t_m) * (k + k)));
	elseif (t_m <= 5.7e+61)
		tmp = ((l * (l / (k * k))) / (t_m * t_m)) / t_m;
	else
		tmp = t_2 * (1.0 / (((t_m / l) * ((t_m / l) * t_m)) * (2.0 * 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_] := Block[{t$95$2 = N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.7e-194], N[(t$95$2 / N[(t$95$m * N[(N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.7e+61], N[(N[(N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], N[(t$95$2 * N[(1.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3.70000000000000008e-194

   1. Initial program 53.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2 \cdot 1}}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\sin k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{2}{\sin k} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{2}{\sin k} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2}{\sin k}} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   3. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\frac{2}{\sin k} \cdot \frac{1}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \tan k\right)}} \]

   4. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 55.4

         \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \]

   6. Applied rewrites 55.4%

      \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\sin k} \cdot \frac{1}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(2 \cdot k\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\sin k} \cdot \color{blue}{\frac{1}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(2 \cdot k\right)}} \]

      3. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\sin k}}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(2 \cdot k\right)}} \]

      4. lower-/.f64 55.4

         \[\leadsto \color{blue}{\frac{\frac{2}{\sin k}}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(2 \cdot k\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\sin k}}{\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(2 \cdot k\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\sin k}}{\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right)} \cdot \left(2 \cdot k\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\sin k}}{\left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(2 \cdot k\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\frac{2}{\sin k}}{\color{blue}{\left(t \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right)} \cdot \left(2 \cdot k\right)} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\frac{2}{\sin k}}{\color{blue}{t \cdot \left(\left(t \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(2 \cdot k\right)\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\sin k}}{\color{blue}{t \cdot \left(\left(t \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(2 \cdot k\right)\right)}} \]

   8. Applied rewrites 60.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\sin k}}{t \cdot \left(\left(\frac{t}{\ell \cdot \ell} \cdot t\right) \cdot \left(k + k\right)\right)}} \]

   if 3.70000000000000008e-194 < t < 5.70000000000000021e61

   1. Initial program 53.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.6

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{\color{blue}{{t}^{3}}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{{t}^{\color{blue}{3}}} \]

      8. unpow3 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{\left(t \cdot t\right) \cdot \color{blue}{t}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{\left(t \cdot t\right) \cdot t} \]

      10. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot t}}{\color{blue}{t}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot t}}{\color{blue}{t}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot t}}{t} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot t}}{t} \]

      14. associate-/l* N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t \cdot t}}{t} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t \cdot t}}{t} \]

      16. lower-/.f64 58.0

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t \cdot t}}{t} \]

      17. lift-pow.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t \cdot t}}{t} \]

      18. unpow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t \cdot t}}{t} \]

      19. lower-*.f64 58.0

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t \cdot t}}{t} \]

   6. Applied rewrites 58.0%

      \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t \cdot t}}{\color{blue}{t}} \]

   if 5.70000000000000021e61 < t

   1. Initial program 53.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2 \cdot 1}}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\sin k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{2}{\sin k} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{2}{\sin k} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2}{\sin k}} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   3. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\frac{2}{\sin k} \cdot \frac{1}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \tan k\right)}} \]

   4. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 55.4

         \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \]

   6. Applied rewrites 55.4%

      \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right)} \cdot \left(2 \cdot k\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\color{blue}{\left(\frac{t}{\ell \cdot \ell} \cdot \left(t \cdot t\right)\right)} \cdot \left(2 \cdot k\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\left(\color{blue}{\frac{t}{\ell \cdot \ell}} \cdot \left(t \cdot t\right)\right) \cdot \left(2 \cdot k\right)} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\color{blue}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell}} \cdot \left(2 \cdot k\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(2 \cdot k\right)} \]

      6. times-frac N/A

         \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)} \cdot \left(2 \cdot k\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t \cdot t}{\ell}\right) \cdot \left(2 \cdot k\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)} \cdot \left(2 \cdot k\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{t \cdot t}}{\ell}\right) \cdot \left(2 \cdot k\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}\right) \cdot \left(2 \cdot k\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\left(\frac{t}{\ell} \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot t\right)\right) \cdot \left(2 \cdot k\right)} \]

      12. lower-*.f64 62.1

          \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}\right) \cdot \left(2 \cdot k\right)} \]

   8. Applied rewrites 62.1%

      \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot t\right)\right)} \cdot \left(2 \cdot k\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 64.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.7 \cdot 10^{-194}:\\ \;\;\;\;\frac{\frac{2}{\sin k}}{t\_m \cdot \left(\left(\frac{t\_m}{\ell \cdot \ell} \cdot t\_m\right) \cdot \left(k + k\right)\right)}\\ \mathbf{elif}\;t\_m \leq 20:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t\_m \cdot t\_m}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t\_m \cdot \left(t\_m \cdot \left(k \cdot t\_m\right)\right)\right) \cdot k} \cdot \ell\\ \end{array} \end{array} \]

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 3.7e-194)
    (/ (/ 2.0 (sin k)) (* t_m (* (* (/ t_m (* l l)) t_m) (+ k k))))
    (if (<= t_m 20.0)
      (/ (/ (* l (/ l (* k k))) (* t_m t_m)) t_m)
      (* (/ l (* (* t_m (* t_m (* k t_m))) k)) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.7e-194) {
		tmp = (2.0 / sin(k)) / (t_m * (((t_m / (l * l)) * t_m) * (k + k)));
	} else if (t_m <= 20.0) {
		tmp = ((l * (l / (k * k))) / (t_m * t_m)) / t_m;
	} else {
		tmp = (l / ((t_m * (t_m * (k * t_m))) * 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 <= 3.7d-194) then
        tmp = (2.0d0 / sin(k)) / (t_m * (((t_m / (l * l)) * t_m) * (k + k)))
    else if (t_m <= 20.0d0) then
        tmp = ((l * (l / (k * k))) / (t_m * t_m)) / t_m
    else
        tmp = (l / ((t_m * (t_m * (k * t_m))) * 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 <= 3.7e-194) {
		tmp = (2.0 / Math.sin(k)) / (t_m * (((t_m / (l * l)) * t_m) * (k + k)));
	} else if (t_m <= 20.0) {
		tmp = ((l * (l / (k * k))) / (t_m * t_m)) / t_m;
	} else {
		tmp = (l / ((t_m * (t_m * (k * t_m))) * 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 <= 3.7e-194:
		tmp = (2.0 / math.sin(k)) / (t_m * (((t_m / (l * l)) * t_m) * (k + k)))
	elif t_m <= 20.0:
		tmp = ((l * (l / (k * k))) / (t_m * t_m)) / t_m
	else:
		tmp = (l / ((t_m * (t_m * (k * t_m))) * k)) * l
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.7e-194)
		tmp = Float64(Float64(2.0 / sin(k)) / Float64(t_m * Float64(Float64(Float64(t_m / Float64(l * l)) * t_m) * Float64(k + k))));
	elseif (t_m <= 20.0)
		tmp = Float64(Float64(Float64(l * Float64(l / Float64(k * k))) / Float64(t_m * t_m)) / t_m);
	else
		tmp = Float64(Float64(l / Float64(Float64(t_m * Float64(t_m * Float64(k * t_m))) * 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 <= 3.7e-194)
		tmp = (2.0 / sin(k)) / (t_m * (((t_m / (l * l)) * t_m) * (k + k)));
	elseif (t_m <= 20.0)
		tmp = ((l * (l / (k * k))) / (t_m * t_m)) / t_m;
	else
		tmp = (l / ((t_m * (t_m * (k * t_m))) * 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, 3.7e-194], N[(N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 20.0], N[(N[(N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], N[(N[(l / N[(N[(t$95$m * N[(t$95$m * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.70000000000000008e-194

   1. Initial program 53.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2 \cdot 1}}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\sin k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{2}{\sin k} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{2}{\sin k} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2}{\sin k}} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   3. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\frac{2}{\sin k} \cdot \frac{1}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \tan k\right)}} \]

   4. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 55.4

         \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(2 \cdot \color{blue}{k}\right)} \]

   6. Applied rewrites 55.4%

      \[\leadsto \frac{2}{\sin k} \cdot \frac{1}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\sin k} \cdot \frac{1}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(2 \cdot k\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\sin k} \cdot \color{blue}{\frac{1}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(2 \cdot k\right)}} \]

      3. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\sin k}}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(2 \cdot k\right)}} \]

      4. lower-/.f64 55.4

         \[\leadsto \color{blue}{\frac{\frac{2}{\sin k}}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(2 \cdot k\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\sin k}}{\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(2 \cdot k\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\sin k}}{\color{blue}{\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right)} \cdot \left(2 \cdot k\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\sin k}}{\left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(2 \cdot k\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\frac{2}{\sin k}}{\color{blue}{\left(t \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right)} \cdot \left(2 \cdot k\right)} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\frac{2}{\sin k}}{\color{blue}{t \cdot \left(\left(t \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(2 \cdot k\right)\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\sin k}}{\color{blue}{t \cdot \left(\left(t \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(2 \cdot k\right)\right)}} \]

   8. Applied rewrites 60.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\sin k}}{t \cdot \left(\left(\frac{t}{\ell \cdot \ell} \cdot t\right) \cdot \left(k + k\right)\right)}} \]

   if 3.70000000000000008e-194 < t < 20

   1. Initial program 53.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.6

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{\color{blue}{{t}^{3}}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{{t}^{\color{blue}{3}}} \]

      8. unpow3 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{\left(t \cdot t\right) \cdot \color{blue}{t}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{\left(t \cdot t\right) \cdot t} \]

      10. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot t}}{\color{blue}{t}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot t}}{\color{blue}{t}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot t}}{t} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot t}}{t} \]

      14. associate-/l* N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t \cdot t}}{t} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t \cdot t}}{t} \]

      16. lower-/.f64 58.0

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t \cdot t}}{t} \]

      17. lift-pow.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t \cdot t}}{t} \]

      18. unpow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t \cdot t}}{t} \]

      19. lower-*.f64 58.0

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t \cdot t}}{t} \]

   6. Applied rewrites 58.0%

      \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t \cdot t}}{\color{blue}{t}} \]

   if 20 < t

   1. Initial program 53.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.6

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.3

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.2

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. unpow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lower-*.f64 59.2

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.2%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 59.2

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   8. Applied rewrites 59.2%

      \[\leadsto \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k} \cdot \ell \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(k \cdot t\right)\right)\right) \cdot k} \cdot \ell \]

      9. lower-*.f64 63.1

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(k \cdot t\right)\right)\right) \cdot k} \cdot \ell \]

   10. Applied rewrites 63.1%

       \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(k \cdot t\right)\right)\right) \cdot k} \cdot \ell \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 63.3% accurate, 5.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{-118}:\\ \;\;\;\;\frac{\ell}{\left(t\_m \cdot \left(t\_m \cdot \left(k \cdot t\_m\right)\right)\right) \cdot k} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t\_m}}{t\_m \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.4e-118)
    (* (/ l (* (* t_m (* t_m (* k t_m))) k)) l)
    (/ (/ (* l (/ l (* k k))) t_m) (* 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 <= 1.4e-118) {
		tmp = (l / ((t_m * (t_m * (k * t_m))) * k)) * l;
	} else {
		tmp = ((l * (l / (k * k))) / t_m) / (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 <= 1.4d-118) then
        tmp = (l / ((t_m * (t_m * (k * t_m))) * k)) * l
    else
        tmp = ((l * (l / (k * k))) / t_m) / (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 <= 1.4e-118) {
		tmp = (l / ((t_m * (t_m * (k * t_m))) * k)) * l;
	} else {
		tmp = ((l * (l / (k * k))) / t_m) / (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 <= 1.4e-118:
		tmp = (l / ((t_m * (t_m * (k * t_m))) * k)) * l
	else:
		tmp = ((l * (l / (k * k))) / t_m) / (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 <= 1.4e-118)
		tmp = Float64(Float64(l / Float64(Float64(t_m * Float64(t_m * Float64(k * t_m))) * k)) * l);
	else
		tmp = Float64(Float64(Float64(l * Float64(l / Float64(k * k))) / t_m) / Float64(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 <= 1.4e-118)
		tmp = (l / ((t_m * (t_m * (k * t_m))) * k)) * l;
	else
		tmp = ((l * (l / (k * k))) / t_m) / (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, 1.4e-118], N[(N[(l / N[(N[(t$95$m * N[(t$95$m * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.4e-118

   1. Initial program 53.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.6

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.3

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.2

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. unpow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lower-*.f64 59.2

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.2%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 59.2

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   8. Applied rewrites 59.2%

      \[\leadsto \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k} \cdot \ell \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(k \cdot t\right)\right)\right) \cdot k} \cdot \ell \]

      9. lower-*.f64 63.1

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(k \cdot t\right)\right)\right) \cdot k} \cdot \ell \]

   10. Applied rewrites 63.1%

       \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(k \cdot t\right)\right)\right) \cdot k} \cdot \ell \]

   if 1.4e-118 < k

   1. Initial program 53.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.6

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{\color{blue}{{t}^{3}}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{{t}^{\color{blue}{3}}} \]

      8. cube-mult N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot \color{blue}{\left(t \cdot t\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot \left(t \cdot \color{blue}{t}\right)} \]

      10. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t}}{\color{blue}{t \cdot t}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t}}{\color{blue}{t \cdot t}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t}}{\color{blue}{t} \cdot t} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t}}{t \cdot t} \]

      14. associate-/l* N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t}}{t \cdot t} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t}}{t \cdot t} \]

      16. lower-/.f64 58.0

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t}}{t \cdot t} \]

      17. lift-pow.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t}}{t \cdot t} \]

      18. unpow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t}}{t \cdot t} \]

      19. lower-*.f64 58.0

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t}}{t \cdot t} \]

   6. Applied rewrites 58.0%

      \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t}}{\color{blue}{t \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 63.1% accurate, 6.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\ell}{\left(t\_m \cdot \left(t\_m \cdot \left(k \cdot t\_m\right)\right)\right) \cdot k} \cdot \ell\right) \end{array} \]

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 (* (* t_m (* t_m (* k t_m))) k)) l)))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / ((t_m * (t_m * (k * t_m))) * k)) * l);
}

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 * (k * t_m))) * k)) * l)
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 * (k * t_m))) * k)) * l);
}

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 * (k * t_m))) * k)) * l)

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 / Float64(Float64(t_m * Float64(t_m * Float64(k * t_m))) * k)) * l))
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 * (k * t_m))) * k)) * l);
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 / N[(N[(t$95$m * N[(t$95$m * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   4. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   5. lower-pow.f64 50.6

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 50.6%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   6. lower-/.f64 55.3

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   8. *-commutative N/A

      \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

   9. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

   10. unpow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

   11. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

   12. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

   13. lower-*.f64 59.2

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   14. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   15. unpow3 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   16. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   17. lower-*.f64 59.2

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

6. Applied rewrites 59.2%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   3. lower-*.f64 59.2

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

8. Applied rewrites 59.2%

   \[\leadsto \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   3. associate-*l* N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k} \cdot \ell \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k} \cdot \ell \]

   5. associate-*l* N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

   8. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(k \cdot t\right)\right)\right) \cdot k} \cdot \ell \]

   9. lower-*.f64 63.1

      \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(k \cdot t\right)\right)\right) \cdot k} \cdot \ell \]

10. Applied rewrites 63.1%

    \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(k \cdot t\right)\right)\right) \cdot k} \cdot \ell \]
11. Add Preprocessing

Alternative 15: 61.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\ell}{\left(t\_m \cdot t\_m\right) \cdot \left(\left(k \cdot t\_m\right) \cdot k\right)} \cdot \ell\right) \end{array} \]

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 (* (* t_m t_m) (* (* k t_m) k))) l)))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / ((t_m * t_m) * ((k * t_m) * k))) * l);
}

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) * ((k * t_m) * k))) * l)
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) * ((k * t_m) * k))) * l);
}

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) * ((k * t_m) * k))) * l)

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 / Float64(Float64(t_m * t_m) * Float64(Float64(k * t_m) * k))) * l))
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) * ((k * t_m) * k))) * l);
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 / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   4. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   5. lower-pow.f64 50.6

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 50.6%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   6. lower-/.f64 55.3

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   8. *-commutative N/A

      \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

   9. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

   10. unpow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

   11. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

   12. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

   13. lower-*.f64 59.2

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   14. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   15. unpow3 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   16. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   17. lower-*.f64 59.2

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

6. Applied rewrites 59.2%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   3. lower-*.f64 59.2

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

8. Applied rewrites 59.2%

   \[\leadsto \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right)} \cdot \ell \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right)} \cdot \ell \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right)} \cdot \ell \]

   7. pow2 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left({t}^{2} \cdot t\right)\right)} \cdot \ell \]

   8. exp-to-pow N/A

      \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(e^{\log t \cdot 2} \cdot t\right)\right)} \cdot \ell \]

   9. lift-log.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(e^{\log t \cdot 2} \cdot t\right)\right)} \cdot \ell \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(e^{\log t \cdot 2} \cdot t\right)\right)} \cdot \ell \]

   11. lift-exp.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(e^{\log t \cdot 2} \cdot t\right)\right)} \cdot \ell \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(e^{\log t \cdot 2} \cdot t\right)\right)} \cdot \ell \]

   13. *-commutative N/A

       \[\leadsto \frac{\ell}{k \cdot \left(\left(e^{\log t \cdot 2} \cdot t\right) \cdot k\right)} \cdot \ell \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(\left(e^{\log t \cdot 2} \cdot t\right) \cdot k\right)} \cdot \ell \]

   15. *-commutative N/A

       \[\leadsto \frac{\ell}{\left(\left(e^{\log t \cdot 2} \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

10. Applied rewrites 61.2%

    \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot t\right) \cdot k\right)} \cdot \ell \]
11. Add Preprocessing

Alternative 16: 58.1% accurate, 6.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\ell}{t\_m \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell\right) \end{array} \]

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 (* t_m (* (* t_m t_m) (* k 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) {
	return t_s * ((l / (t_m * ((t_m * t_m) * (k * k)))) * l);
}

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 * t_m) * (k * k)))) * l)
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 * t_m) * (k * k)))) * l);
}

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 * t_m) * (k * k)))) * l)

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 / Float64(t_m * Float64(Float64(t_m * t_m) * Float64(k * k)))) * l))
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 * t_m) * (k * k)))) * l);
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 / N[(t$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   4. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   5. lower-pow.f64 50.6

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 50.6%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   6. lower-/.f64 55.3

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   8. *-commutative N/A

      \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

   9. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

   10. unpow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

   11. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

   12. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

   13. lower-*.f64 59.2

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   14. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   15. unpow3 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   16. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   17. lower-*.f64 59.2

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

6. Applied rewrites 59.2%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   3. lower-*.f64 59.2

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

8. Applied rewrites 59.2%

   \[\leadsto \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right)} \cdot \ell \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right)} \cdot \ell \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right)} \cdot \ell \]

   7. pow2 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left({t}^{2} \cdot t\right)\right)} \cdot \ell \]

   8. exp-to-pow N/A

      \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(e^{\log t \cdot 2} \cdot t\right)\right)} \cdot \ell \]

   9. lift-log.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(e^{\log t \cdot 2} \cdot t\right)\right)} \cdot \ell \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(e^{\log t \cdot 2} \cdot t\right)\right)} \cdot \ell \]

   11. lift-exp.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(e^{\log t \cdot 2} \cdot t\right)\right)} \cdot \ell \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(e^{\log t \cdot 2} \cdot t\right)\right)} \cdot \ell \]

   13. *-commutative N/A

       \[\leadsto \frac{\ell}{k \cdot \left(\left(e^{\log t \cdot 2} \cdot t\right) \cdot k\right)} \cdot \ell \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(\left(e^{\log t \cdot 2} \cdot t\right) \cdot k\right)} \cdot \ell \]

   15. *-commutative N/A

       \[\leadsto \frac{\ell}{\left(\left(e^{\log t \cdot 2} \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(\left(e^{\log t \cdot 2} \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   17. associate-*l* N/A

       \[\leadsto \frac{\ell}{\left(e^{\log t \cdot 2} \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \ell \]

10. Applied rewrites 58.1%

    \[\leadsto \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025154 
(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))))