Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.8% → 91.3%

Time: 6.9s

Alternatives: 18

Speedup: 6.6×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.8% 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: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.05 \cdot 10^{-80}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\_m\right) \cdot k\right) \cdot k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\tan k \cdot t\_m}{\ell} \cdot \left(\frac{\sin k \cdot t\_m}{\ell} \cdot t\_m\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \]

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 1.05e-80)
    (*
     2.0
     (* l (* l (/ (cos k) (* (* (* (- 0.5 (* 0.5 (cos (+ k k)))) t_m) k) k)))))
    (/
     2.0
     (*
      (* (/ (* (tan k) t_m) l) (* (/ (* (sin k) t_m) l) t_m))
      (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.05e-80) {
		tmp = 2.0 * (l * (l * (cos(k) / ((((0.5 - (0.5 * cos((k + k)))) * t_m) * k) * k))));
	} else {
		tmp = 2.0 / ((((tan(k) * t_m) / l) * (((sin(k) * t_m) / l) * t_m)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.05d-80) then
        tmp = 2.0d0 * (l * (l * (cos(k) / ((((0.5d0 - (0.5d0 * cos((k + k)))) * t_m) * k) * k))))
    else
        tmp = 2.0d0 / ((((tan(k) * t_m) / l) * (((sin(k) * t_m) / l) * t_m)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.05e-80) {
		tmp = 2.0 * (l * (l * (Math.cos(k) / ((((0.5 - (0.5 * Math.cos((k + k)))) * t_m) * k) * k))));
	} else {
		tmp = 2.0 / ((((Math.tan(k) * t_m) / l) * (((Math.sin(k) * t_m) / l) * t_m)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.05e-80:
		tmp = 2.0 * (l * (l * (math.cos(k) / ((((0.5 - (0.5 * math.cos((k + k)))) * t_m) * k) * k))))
	else:
		tmp = 2.0 / ((((math.tan(k) * t_m) / l) * (((math.sin(k) * t_m) / l) * t_m)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.05e-80)
		tmp = Float64(2.0 * Float64(l * Float64(l * Float64(cos(k) / Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))) * t_m) * k) * k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k) * t_m) / l) * Float64(Float64(Float64(sin(k) * t_m) / l) * t_m)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.05e-80)
		tmp = 2.0 * (l * (l * (cos(k) / ((((0.5 - (0.5 * cos((k + k)))) * t_m) * k) * k))));
	else
		tmp = 2.0 / ((((tan(k) * t_m) / l) * (((sin(k) * t_m) / l) * t_m)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.05e-80], N[(2.0 * N[(l * N[(l * N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.05000000000000001e-80

   1. Initial program 54.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 64.8

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*r* N/A

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

   6. Applied rewrites 65.5%

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}\right)}\right) \]

   if 1.05000000000000001e-80 < t

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 67.6

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

   3. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.5

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 78.4

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

   7. Applied rewrites 78.4%

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

Alternative 2: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.05 \cdot 10^{-80}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\_m\right) \cdot k\right) \cdot k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(\left(\tan k \cdot t\_m\right) \cdot \frac{\sin k \cdot t\_m}{\ell}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \]

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 1.05e-80)
    (*
     2.0
     (* l (* l (/ (cos k) (* (* (* (- 0.5 (* 0.5 (cos (+ k k)))) t_m) k) k)))))
    (/
     2.0
     (*
      (* (/ t_m l) (* (* (tan k) t_m) (/ (* (sin k) t_m) l)))
      (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.05e-80) {
		tmp = 2.0 * (l * (l * (cos(k) / ((((0.5 - (0.5 * cos((k + k)))) * t_m) * k) * k))));
	} else {
		tmp = 2.0 / (((t_m / l) * ((tan(k) * t_m) * ((sin(k) * t_m) / l))) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.05d-80) then
        tmp = 2.0d0 * (l * (l * (cos(k) / ((((0.5d0 - (0.5d0 * cos((k + k)))) * t_m) * k) * k))))
    else
        tmp = 2.0d0 / (((t_m / l) * ((tan(k) * t_m) * ((sin(k) * t_m) / l))) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.05e-80) {
		tmp = 2.0 * (l * (l * (Math.cos(k) / ((((0.5 - (0.5 * Math.cos((k + k)))) * t_m) * k) * k))));
	} else {
		tmp = 2.0 / (((t_m / l) * ((Math.tan(k) * t_m) * ((Math.sin(k) * t_m) / l))) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.05e-80:
		tmp = 2.0 * (l * (l * (math.cos(k) / ((((0.5 - (0.5 * math.cos((k + k)))) * t_m) * k) * k))))
	else:
		tmp = 2.0 / (((t_m / l) * ((math.tan(k) * t_m) * ((math.sin(k) * t_m) / l))) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.05e-80)
		tmp = Float64(2.0 * Float64(l * Float64(l * Float64(cos(k) / Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))) * t_m) * k) * k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(Float64(tan(k) * t_m) * Float64(Float64(sin(k) * t_m) / l))) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.05e-80)
		tmp = 2.0 * (l * (l * (cos(k) / ((((0.5 - (0.5 * cos((k + k)))) * t_m) * k) * k))));
	else
		tmp = 2.0 / (((t_m / l) * ((tan(k) * t_m) * ((sin(k) * t_m) / l))) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.05e-80], N[(2.0 * N[(l * N[(l * N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.05000000000000001e-80

   1. Initial program 54.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 64.8

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*r* N/A

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

   6. Applied rewrites 65.5%

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}\right)}\right) \]

   if 1.05000000000000001e-80 < t

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 67.6

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

   3. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.5

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 77.2

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

   7. Applied rewrites 77.2%

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

Alternative 3: 90.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sin k \cdot t\_m\\ t_3 := \frac{\tan k \cdot t\_m}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.05 \cdot 10^{-80}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\_m\right) \cdot k\right) \cdot k}\right)\right)\\ \mathbf{elif}\;t\_m \leq 7 \cdot 10^{+141}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right) \cdot \left(\frac{t\_2 \cdot t\_m}{\ell} \cdot t\_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_3 \cdot \left(\frac{t\_2}{\ell} \cdot t\_m\right)\right) \cdot 2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (sin k) t_m)) (t_3 (/ (* (tan k) t_m) l)))
   (*
    t_s
    (if (<= t_m 1.05e-80)
      (*
       2.0
       (*
        l
        (* l (/ (cos k) (* (* (* (- 0.5 (* 0.5 (cos (+ k k)))) t_m) k) k)))))
      (if (<= t_m 7e+141)
        (/ 2.0 (* (fma k (/ k (* t_m t_m)) 2.0) (* (/ (* t_2 t_m) l) t_3)))
        (/ 2.0 (* (* t_3 (* (/ t_2 l) 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;
	double t_3 = (tan(k) * t_m) / l;
	double tmp;
	if (t_m <= 1.05e-80) {
		tmp = 2.0 * (l * (l * (cos(k) / ((((0.5 - (0.5 * cos((k + k)))) * t_m) * k) * k))));
	} else if (t_m <= 7e+141) {
		tmp = 2.0 / (fma(k, (k / (t_m * t_m)), 2.0) * (((t_2 * t_m) / l) * t_3));
	} else {
		tmp = 2.0 / ((t_3 * ((t_2 / l) * t_m)) * 2.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(sin(k) * t_m)
	t_3 = Float64(Float64(tan(k) * t_m) / l)
	tmp = 0.0
	if (t_m <= 1.05e-80)
		tmp = Float64(2.0 * Float64(l * Float64(l * Float64(cos(k) / Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))) * t_m) * k) * k)))));
	elseif (t_m <= 7e+141)
		tmp = Float64(2.0 / Float64(fma(k, Float64(k / Float64(t_m * t_m)), 2.0) * Float64(Float64(Float64(t_2 * t_m) / l) * t_3)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_3 * Float64(Float64(t_2 / l) * t_m)) * 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.05e-80], N[(2.0 * N[(l * N[(l * N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7e+141], N[(2.0 / N[(N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[(t$95$2 * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$3 * N[(N[(t$95$2 / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.05000000000000001e-80

   1. Initial program 54.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 64.8

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*r* N/A

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

   6. Applied rewrites 65.5%

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}\right)}\right) \]

   if 1.05000000000000001e-80 < t < 6.9999999999999999e141

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 67.6

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

   3. Applied rewrites 67.6%

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

   5. Applied rewrites 50.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 63.2

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

   7. Applied rewrites 63.2%

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

   if 6.9999999999999999e141 < t

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 67.6

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

   3. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.5

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 78.4

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

   7. Applied rewrites 78.4%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 70.7%

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

Alternative 4: 78.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(\frac{\tan k \cdot t\_m}{\ell} \cdot \left(\frac{\sin k \cdot t\_m}{\ell} \cdot t\_m\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\_m\right) \cdot k\right) \cdot k}\right)\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.4e-18)
    (/ 2.0 (* (* (/ (* (tan k) t_m) l) (* (/ (* (sin k) t_m) l) t_m)) 2.0))
    (*
     2.0
     (*
      l
      (* l (/ (cos k) (* (* (* (- 0.5 (* 0.5 (cos (+ k k)))) t_m) k) k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.4e-18) {
		tmp = 2.0 / ((((tan(k) * t_m) / l) * (((sin(k) * t_m) / l) * t_m)) * 2.0);
	} else {
		tmp = 2.0 * (l * (l * (cos(k) / ((((0.5 - (0.5 * cos((k + k)))) * t_m) * k) * k))));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.4d-18) then
        tmp = 2.0d0 / ((((tan(k) * t_m) / l) * (((sin(k) * t_m) / l) * t_m)) * 2.0d0)
    else
        tmp = 2.0d0 * (l * (l * (cos(k) / ((((0.5d0 - (0.5d0 * cos((k + k)))) * t_m) * k) * k))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.4e-18) {
		tmp = 2.0 / ((((Math.tan(k) * t_m) / l) * (((Math.sin(k) * t_m) / l) * t_m)) * 2.0);
	} else {
		tmp = 2.0 * (l * (l * (Math.cos(k) / ((((0.5 - (0.5 * Math.cos((k + k)))) * t_m) * k) * k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 3.4e-18:
		tmp = 2.0 / ((((math.tan(k) * t_m) / l) * (((math.sin(k) * t_m) / l) * t_m)) * 2.0)
	else:
		tmp = 2.0 * (l * (l * (math.cos(k) / ((((0.5 - (0.5 * math.cos((k + k)))) * t_m) * k) * k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3.4e-18)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k) * t_m) / l) * Float64(Float64(Float64(sin(k) * t_m) / l) * t_m)) * 2.0));
	else
		tmp = Float64(2.0 * Float64(l * Float64(l * Float64(cos(k) / Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))) * t_m) * k) * k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 3.4e-18)
		tmp = 2.0 / ((((tan(k) * t_m) / l) * (((sin(k) * t_m) / l) * t_m)) * 2.0);
	else
		tmp = 2.0 * (l * (l * (cos(k) / ((((0.5 - (0.5 * cos((k + k)))) * t_m) * k) * k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 3.4e-18], N[(2.0 / N[(N[(N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(l * N[(l * N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.40000000000000001e-18

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 67.6

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

   3. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.5

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 78.4

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

   7. Applied rewrites 78.4%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 70.7%

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

   if 3.40000000000000001e-18 < k

   1. Initial program 54.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 64.8

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*r* N/A

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

   6. Applied rewrites 65.5%

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 77.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.3 \cdot 10^{-19}:\\ \;\;\;\;\frac{2}{\left(\frac{\tan k \cdot t\_m}{\ell} \cdot \left(\frac{\sin k \cdot t\_m}{\ell} \cdot t\_m\right)\right) \cdot 2}\\ \mathbf{elif}\;k \leq 0.0092:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\frac{\ell}{\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot k\right) \cdot k} \cdot \frac{\cos k}{\left(k \cdot k\right) \cdot t\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\_m\right) \cdot k\right) \cdot k}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 5.3e-19)
    (/ 2.0 (* (* (/ (* (tan k) t_m) l) (* (/ (* (sin k) t_m) l) t_m)) 2.0))
    (if (<= k 0.0092)
      (*
       2.0
       (*
        l
        (*
         (/ l (* (* (fma -0.3333333333333333 (* k k) 1.0) k) k))
         (/ (cos k) (* (* k k) t_m)))))
      (*
       2.0
       (*
        (cos k)
        (/ (* l l) (* (* (* (- 0.5 (* 0.5 (cos (+ k k)))) t_m) k) k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5.3e-19) {
		tmp = 2.0 / ((((tan(k) * t_m) / l) * (((sin(k) * t_m) / l) * t_m)) * 2.0);
	} else if (k <= 0.0092) {
		tmp = 2.0 * (l * ((l / ((fma(-0.3333333333333333, (k * k), 1.0) * k) * k)) * (cos(k) / ((k * k) * t_m))));
	} else {
		tmp = 2.0 * (cos(k) * ((l * l) / ((((0.5 - (0.5 * cos((k + k)))) * t_m) * k) * k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 5.3e-19)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k) * t_m) / l) * Float64(Float64(Float64(sin(k) * t_m) / l) * t_m)) * 2.0));
	elseif (k <= 0.0092)
		tmp = Float64(2.0 * Float64(l * Float64(Float64(l / Float64(Float64(fma(-0.3333333333333333, Float64(k * k), 1.0) * k) * k)) * Float64(cos(k) / Float64(Float64(k * k) * t_m)))));
	else
		tmp = Float64(2.0 * Float64(cos(k) * Float64(Float64(l * l) / Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))) * t_m) * k) * k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 5.3e-19], N[(2.0 / N[(N[(N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.0092], N[(2.0 * N[(l * N[(N[(l / N[(N[(N[(-0.3333333333333333 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[k], $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 5.29999999999999972e-19

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 67.6

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

   3. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.5

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 78.4

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

   7. Applied rewrites 78.4%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 70.7%

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

   if 5.29999999999999972e-19 < k < 0.0091999999999999998

   1. Initial program 54.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 61.2

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lift-sin.f64 N/A

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

      21. lift-sin.f64 N/A

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

      22. sqr-sin-a N/A

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

   6. Applied rewrites 56.9%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}\right) \]

   7. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 52.4

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(1 + -0.3333333333333333 \cdot {k}^{2}\right)}\right) \]

   9. Applied rewrites 52.4%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{\left(1 + -0.3333333333333333 \cdot {k}^{2}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-/l* N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

   11. Applied rewrites 55.6%

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

   if 0.0091999999999999998 < k

   1. Initial program 54.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 60.0

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 59.4%

      \[\leadsto 2 \cdot \left(\cos k \cdot \color{blue}{\frac{\ell \cdot \ell}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 77.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(\frac{\tan k \cdot t\_m}{\ell} \cdot \left(\frac{\sin k \cdot t\_m}{\ell} \cdot t\_m\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2 \cdot \cos k}{\left(k \cdot k\right) \cdot t\_m} \cdot \ell\right) \cdot \frac{\ell}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}\\ \end{array} \end{array} \]

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 2.3e-18)
    (/ 2.0 (* (* (/ (* (tan k) t_m) l) (* (/ (* (sin k) t_m) l) t_m)) 2.0))
    (*
     (* (/ (* 2.0 (cos k)) (* (* k k) t_m)) l)
     (/ l (fma (cos (+ k k)) -0.5 0.5))))))

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 <= 2.3e-18) {
		tmp = 2.0 / ((((tan(k) * t_m) / l) * (((sin(k) * t_m) / l) * t_m)) * 2.0);
	} else {
		tmp = (((2.0 * cos(k)) / ((k * k) * t_m)) * l) * (l / fma(cos((k + k)), -0.5, 0.5));
	}
	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 <= 2.3e-18)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k) * t_m) / l) * Float64(Float64(Float64(sin(k) * t_m) / l) * t_m)) * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 * cos(k)) / Float64(Float64(k * k) * t_m)) * l) * Float64(l / fma(cos(Float64(k + k)), -0.5, 0.5)));
	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, 2.3e-18], N[(2.0 / N[(N[(N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * N[(l / N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.3000000000000001e-18

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 67.6

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

   3. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.5

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 78.4

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

   7. Applied rewrites 78.4%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 70.7%

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

   if 2.3000000000000001e-18 < k

   1. Initial program 54.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 61.2

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lift-sin.f64 N/A

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

      21. lift-sin.f64 N/A

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

      22. sqr-sin-a N/A

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

   6. Applied rewrites 56.9%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   8. Applied rewrites 62.2%

      \[\leadsto \left(\frac{2 \cdot \cos k}{\left(k \cdot k\right) \cdot t} \cdot \ell\right) \cdot \color{blue}{\frac{\ell}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 75.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(k \cdot k\right) \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.3 \cdot 10^{-19}:\\ \;\;\;\;\frac{2}{\left(\frac{\tan k \cdot t\_m}{\ell} \cdot \left(\frac{\sin k \cdot t\_m}{\ell} \cdot t\_m\right)\right) \cdot 2}\\ \mathbf{elif}\;k \leq 0.0092:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\frac{\ell}{\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot k\right) \cdot k} \cdot \frac{\cos k}{t\_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\_2}\right)\\ \end{array} \end{array} \end{array} \]

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 (* (* k k) t_m)))
   (*
    t_s
    (if (<= k 5.3e-19)
      (/ 2.0 (* (* (/ (* (tan k) t_m) l) (* (/ (* (sin k) t_m) l) t_m)) 2.0))
      (if (<= k 0.0092)
        (*
         2.0
         (*
          l
          (*
           (/ l (* (* (fma -0.3333333333333333 (* k k) 1.0) k) k))
           (/ (cos k) t_2))))
        (*
         2.0
         (* (cos k) (/ (* l l) (* (fma (cos (+ k k)) -0.5 0.5) t_2)))))))))

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 = (k * k) * t_m;
	double tmp;
	if (k <= 5.3e-19) {
		tmp = 2.0 / ((((tan(k) * t_m) / l) * (((sin(k) * t_m) / l) * t_m)) * 2.0);
	} else if (k <= 0.0092) {
		tmp = 2.0 * (l * ((l / ((fma(-0.3333333333333333, (k * k), 1.0) * k) * k)) * (cos(k) / t_2)));
	} else {
		tmp = 2.0 * (cos(k) * ((l * l) / (fma(cos((k + k)), -0.5, 0.5) * t_2)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(k * k) * t_m)
	tmp = 0.0
	if (k <= 5.3e-19)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k) * t_m) / l) * Float64(Float64(Float64(sin(k) * t_m) / l) * t_m)) * 2.0));
	elseif (k <= 0.0092)
		tmp = Float64(2.0 * Float64(l * Float64(Float64(l / Float64(Float64(fma(-0.3333333333333333, Float64(k * k), 1.0) * k) * k)) * Float64(cos(k) / t_2))));
	else
		tmp = Float64(2.0 * Float64(cos(k) * Float64(Float64(l * l) / Float64(fma(cos(Float64(k + k)), -0.5, 0.5) * t_2))));
	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[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 5.3e-19], N[(2.0 / N[(N[(N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.0092], N[(2.0 * N[(l * N[(N[(l / N[(N[(N[(-0.3333333333333333 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[k], $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 5.29999999999999972e-19

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 67.6

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

   3. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.5

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 78.4

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

   7. Applied rewrites 78.4%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 70.7%

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

   if 5.29999999999999972e-19 < k < 0.0091999999999999998

   1. Initial program 54.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 61.2

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lift-sin.f64 N/A

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

      21. lift-sin.f64 N/A

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

      22. sqr-sin-a N/A

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

   6. Applied rewrites 56.9%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}\right) \]

   7. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 52.4

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(1 + -0.3333333333333333 \cdot {k}^{2}\right)}\right) \]

   9. Applied rewrites 52.4%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{\left(1 + -0.3333333333333333 \cdot {k}^{2}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-/l* N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

   11. Applied rewrites 55.6%

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

   if 0.0091999999999999998 < k

   1. Initial program 54.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 61.2

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lift-sin.f64 N/A

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

      21. lift-sin.f64 N/A

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

      22. sqr-sin-a N/A

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

   6. Applied rewrites 56.9%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 56.9

         \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}}\right) \]

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. metadata-eval 56.9

          \[\leadsto 2 \cdot \left(\cos k \cdot \frac{\ell \cdot \ell}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left(\left(k \cdot \color{blue}{k}\right) \cdot t\right)}\right) \]

   8. Applied rewrites 56.9%

      \[\leadsto 2 \cdot \left(\cos k \cdot \color{blue}{\frac{\ell \cdot \ell}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 75.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.04 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot t\_m, 0.16666666666666666, t\_m\right)}{\ell} \cdot k\right) \cdot k\right) \cdot \left(\frac{t\_m}{\ell} \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)\right)\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\tan k \cdot t\_m}{\ell} \cdot \left(\frac{\sin k \cdot t\_m}{\ell} \cdot t\_m\right)\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 1.04e-17)
    (/
     2.0
     (*
      (*
       (* (* (/ (fma (* (* k k) t_m) 0.16666666666666666 t_m) l) k) k)
       (* (/ t_m l) (fma (/ k t_m) (/ k t_m) 2.0)))
      t_m))
    (/ 2.0 (* (* (/ (* (tan k) t_m) l) (* (/ (* (sin k) t_m) l) 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 (l <= 1.04e-17) {
		tmp = 2.0 / (((((fma(((k * k) * t_m), 0.16666666666666666, t_m) / l) * k) * k) * ((t_m / l) * fma((k / t_m), (k / t_m), 2.0))) * t_m);
	} else {
		tmp = 2.0 / ((((tan(k) * t_m) / l) * (((sin(k) * t_m) / l) * 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 (l <= 1.04e-17)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(fma(Float64(Float64(k * k) * t_m), 0.16666666666666666, t_m) / l) * k) * k) * Float64(Float64(t_m / l) * fma(Float64(k / t_m), Float64(k / t_m), 2.0))) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k) * t_m) / l) * Float64(Float64(Float64(sin(k) * t_m) / l) * 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[l, 1.04e-17], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * 0.16666666666666666 + t$95$m), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.04e-17

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 67.6

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

   3. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.5

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 64.0

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

   8. Applied rewrites 64.0%

      \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \mathsf{fma}\left(0.16666666666666666, \frac{{k}^{2} \cdot t}{\ell}, \frac{t}{\ell}\right)\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   10. Applied rewrites 70.7%

       \[\leadsto \color{blue}{\frac{2}{\left(\left(\left(\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, 0.16666666666666666, t\right)}{\ell} \cdot k\right) \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)\right) \cdot t}} \]

   if 1.04e-17 < l

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 67.6

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

   3. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.5

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 78.4

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

   7. Applied rewrites 78.4%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 70.7%

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

Alternative 9: 74.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 4.7 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot t\_m, 0.16666666666666666, t\_m\right)}{\ell} \cdot k\right) \cdot k\right) \cdot \left(\frac{t\_m}{\ell} \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)\right)\right) \cdot t\_m}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+144}:\\ \;\;\;\;\frac{\ell}{\left(\left(t\_m \cdot k\right) \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t\_m}{\ell}\right)\right)\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 4.7e-18)
    (/
     2.0
     (*
      (*
       (* (* (/ (fma (* (* k k) t_m) 0.16666666666666666 t_m) l) k) k)
       (* (/ t_m l) (fma (/ k t_m) (/ k t_m) 2.0)))
      t_m))
    (if (<= l 5.8e+144)
      (* (/ l (* (* (* t_m k) t_m) (* k t_m))) l)
      (/
       2.0
       (* (* t_m (* (/ t_m l) (* (tan k) (/ (* (sin k) t_m) l)))) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 4.7e-18) {
		tmp = 2.0 / (((((fma(((k * k) * t_m), 0.16666666666666666, t_m) / l) * k) * k) * ((t_m / l) * fma((k / t_m), (k / t_m), 2.0))) * t_m);
	} else if (l <= 5.8e+144) {
		tmp = (l / (((t_m * k) * t_m) * (k * t_m))) * l;
	} else {
		tmp = 2.0 / ((t_m * ((t_m / l) * (tan(k) * ((sin(k) * t_m) / l)))) * 2.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 4.7e-18)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(fma(Float64(Float64(k * k) * t_m), 0.16666666666666666, t_m) / l) * k) * k) * Float64(Float64(t_m / l) * fma(Float64(k / t_m), Float64(k / t_m), 2.0))) * t_m));
	elseif (l <= 5.8e+144)
		tmp = Float64(Float64(l / Float64(Float64(Float64(t_m * k) * t_m) * Float64(k * t_m))) * l);
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(t_m / l) * Float64(tan(k) * Float64(Float64(sin(k) * t_m) / l)))) * 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 4.7e-18], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * 0.16666666666666666 + t$95$m), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.8e+144], N[(N[(l / N[(N[(N[(t$95$m * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if l < 4.6999999999999996e-18

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. unpow3 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 67.6

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

   3. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.5

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 64.0

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

   8. Applied rewrites 64.0%

      \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \mathsf{fma}\left(0.16666666666666666, \frac{{k}^{2} \cdot t}{\ell}, \frac{t}{\ell}\right)\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   10. Applied rewrites 70.7%

       \[\leadsto \color{blue}{\frac{2}{\left(\left(\left(\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, 0.16666666666666666, t\right)}{\ell} \cdot k\right) \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)\right) \cdot t}} \]

   if 4.6999999999999996e-18 < l < 5.79999999999999996e144

   1. Initial program 54.8%

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

   2. 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 51.1

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

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.7

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

      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 60.1

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

      14. lift-pow.f64 N/A

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

      15. pow3 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. lift-*.f64 60.1

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

   6. Applied rewrites 60.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   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 60.1

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

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      13. lower-*.f64 63.1

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 63.1%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lower-*.f64 66.6

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      7. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      8. lower-*.f64 66.6

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 66.6%

       \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   if 5.79999999999999996e144 < l

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. unpow3 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 67.6

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 67.6%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 74.5

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 74.5

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 74.5%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right)\right) \cdot \color{blue}{2}} \]
   7. Step-by-step derivation

   8. Applied rewrites 67.6%

      \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right)\right) \cdot \color{blue}{2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 72.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\cos k}{\left(k \cdot k\right) \cdot t\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7.4 \cdot 10^{-19}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot t\_m}{\ell} \cdot \left(\frac{\sin k \cdot t\_m}{\ell} \cdot t\_m\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;k \leq 1.65:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\frac{\ell}{\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot k\right) \cdot k} \cdot t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_2 \cdot \frac{\ell \cdot \ell}{0.5 - \left(0.5 + -1 \cdot {k}^{2}\right)}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (cos k) (* (* k k) t_m))))
   (*
    t_s
    (if (<= k 7.4e-19)
      (/
       2.0
       (*
        (* (/ (* k t_m) l) (* (/ (* (sin k) t_m) l) t_m))
        (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
      (if (<= k 1.65)
        (*
         2.0
         (* l (* (/ l (* (* (fma -0.3333333333333333 (* k k) 1.0) k) k)) t_2)))
        (* 2.0 (* t_2 (/ (* l l) (- 0.5 (+ 0.5 (* -1.0 (pow k 2.0))))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = cos(k) / ((k * k) * t_m);
	double tmp;
	if (k <= 7.4e-19) {
		tmp = 2.0 / ((((k * t_m) / l) * (((sin(k) * t_m) / l) * t_m)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else if (k <= 1.65) {
		tmp = 2.0 * (l * ((l / ((fma(-0.3333333333333333, (k * k), 1.0) * k) * k)) * t_2));
	} else {
		tmp = 2.0 * (t_2 * ((l * l) / (0.5 - (0.5 + (-1.0 * pow(k, 2.0))))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(cos(k) / Float64(Float64(k * k) * t_m))
	tmp = 0.0
	if (k <= 7.4e-19)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * t_m) / l) * Float64(Float64(Float64(sin(k) * t_m) / l) * t_m)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	elseif (k <= 1.65)
		tmp = Float64(2.0 * Float64(l * Float64(Float64(l / Float64(Float64(fma(-0.3333333333333333, Float64(k * k), 1.0) * k) * k)) * t_2)));
	else
		tmp = Float64(2.0 * Float64(t_2 * Float64(Float64(l * l) / Float64(0.5 - Float64(0.5 + Float64(-1.0 * (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_] := Block[{t$95$2 = N[(N[Cos[k], $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 7.4e-19], N[(2.0 / N[(N[(N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $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[k, 1.65], N[(2.0 * N[(l * N[(N[(l / N[(N[(N[(-0.3333333333333333 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$2 * N[(N[(l * l), $MachinePrecision] / N[(0.5 - N[(0.5 + N[(-1.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 7.40000000000000011e-19

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. unpow3 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 67.6

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 67.6%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 74.5

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 74.5

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 74.5%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right) \cdot t\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right)} \cdot t\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)}\right) \cdot t\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \frac{\sin k \cdot t}{\ell}\right)} \cdot t\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\color{blue}{\frac{t}{\ell}} \cdot \tan k\right) \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot t\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot \tan k}{\ell}} \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot t\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot \tan k}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot \tan k}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{t \cdot \tan k}{\ell}} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\tan k \cdot t}}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\tan k \cdot t}}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 78.4

          \[\leadsto \frac{2}{\left(\frac{\tan k \cdot t}{\ell} \cdot \color{blue}{\left(\frac{\sin k \cdot t}{\ell} \cdot t\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Applied rewrites 78.4%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{\tan k \cdot t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{\color{blue}{k} \cdot t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 72.0%

       \[\leadsto \frac{2}{\left(\frac{\color{blue}{k} \cdot t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   if 7.40000000000000011e-19 < k < 1.6499999999999999

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 60.0

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 60.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. pow2 N/A

         \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      10. times-frac N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{{\sin k}^{2}}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{{\sin k}^{2}}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{\ell \cdot \color{blue}{\ell}}{{\sin k}^{2}}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right) \]

      17. lower-/.f64 61.2

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\color{blue}{{\sin k}^{2}}}\right) \]

      18. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{\sin k}^{\color{blue}{2}}}\right) \]

      19. unpow2 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \color{blue}{\sin k}}\right) \]

      20. lift-sin.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \sin \color{blue}{k}}\right) \]

      21. lift-sin.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \sin k}\right) \]

      22. sqr-sin-a N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot k\right)}}\right) \]

   6. Applied rewrites 56.9%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}\right) \]

   7. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{\left(1 + \frac{-1}{3} \cdot {k}^{2}\right)}}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(1 + \color{blue}{\frac{-1}{3} \cdot {k}^{2}}\right)}\right) \]

      2. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(1 + \color{blue}{\frac{-1}{3}} \cdot {k}^{2}\right)}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(1 + \frac{-1}{3} \cdot \color{blue}{{k}^{2}}\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {k}^{\color{blue}{2}}\right)}\right) \]

      5. lower-pow.f64 52.4

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(1 + -0.3333333333333333 \cdot {k}^{2}\right)}\right) \]

   9. Applied rewrites 52.4%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{\left(1 + -0.3333333333333333 \cdot {k}^{2}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{{k}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {k}^{2}\right)}}\right) \]

       2. *-commutative N/A

          \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{{k}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {k}^{2}\right)} \cdot \color{blue}{\frac{\cos k}{\left(k \cdot k\right) \cdot t}}\right) \]

       3. lift-/.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{{k}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {k}^{2}\right)} \cdot \frac{\color{blue}{\cos k}}{\left(k \cdot k\right) \cdot t}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{{k}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {k}^{2}\right)} \cdot \frac{\cos \color{blue}{k}}{\left(k \cdot k\right) \cdot t}\right) \]

       5. associate-/l* N/A

          \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{{k}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {k}^{2}\right)}\right) \cdot \frac{\color{blue}{\cos k}}{\left(k \cdot k\right) \cdot t}\right) \]

       6. associate-*l* N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\frac{\ell}{{k}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {k}^{2}\right)} \cdot \frac{\cos k}{\left(k \cdot k\right) \cdot t}\right)}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\frac{\ell}{{k}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {k}^{2}\right)} \cdot \frac{\cos k}{\left(k \cdot k\right) \cdot t}\right)}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\ell \cdot \left(\frac{\ell}{{k}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {k}^{2}\right)} \cdot \color{blue}{\frac{\cos k}{\left(k \cdot k\right) \cdot t}}\right)\right) \]

   11. Applied rewrites 55.6%

       \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\frac{\ell}{\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot k\right) \cdot k} \cdot \frac{\cos k}{\left(k \cdot k\right) \cdot t}\right)}\right) \]

   if 1.6499999999999999 < k

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 60.0

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 60.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. pow2 N/A

         \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      10. times-frac N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{{\sin k}^{2}}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{{\sin k}^{2}}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{\ell \cdot \color{blue}{\ell}}{{\sin k}^{2}}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right) \]

      17. lower-/.f64 61.2

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\color{blue}{{\sin k}^{2}}}\right) \]

      18. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{\sin k}^{\color{blue}{2}}}\right) \]

      19. unpow2 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \color{blue}{\sin k}}\right) \]

      20. lift-sin.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \sin \color{blue}{k}}\right) \]

      21. lift-sin.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \sin k}\right) \]

      22. sqr-sin-a N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot k\right)}}\right) \]

   6. Applied rewrites 56.9%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}\right) \]

   7. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \left(\frac{1}{2} + \color{blue}{-1 \cdot {k}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \left(\frac{1}{2} + -1 \cdot \color{blue}{{k}^{2}}\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \left(\frac{1}{2} + -1 \cdot {k}^{\color{blue}{2}}\right)}\right) \]

      3. lower-pow.f64 51.4

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{0.5 - \left(0.5 + -1 \cdot {k}^{2}\right)}\right) \]

   9. Applied rewrites 51.4%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{0.5 - \left(0.5 + \color{blue}{-1 \cdot {k}^{2}}\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 72.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot t\_m}{\ell} \cdot \left(\frac{\sin k \cdot t\_m}{\ell} \cdot t\_m\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t\_m} \cdot \frac{{\ell}^{2}}{{k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.3e-18)
    (/
     2.0
     (*
      (* (/ (* k t_m) l) (* (/ (* (sin k) t_m) l) t_m))
      (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
    (* 2.0 (* (/ (cos k) (* (* k k) t_m)) (/ (pow l 2.0) (pow 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 (k <= 2.3e-18) {
		tmp = 2.0 / ((((k * t_m) / l) * (((sin(k) * t_m) / l) * t_m)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else {
		tmp = 2.0 * ((cos(k) / ((k * k) * t_m)) * (pow(l, 2.0) / pow(k, 2.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.3d-18) then
        tmp = 2.0d0 / ((((k * t_m) / l) * (((sin(k) * t_m) / l) * t_m)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    else
        tmp = 2.0d0 * ((cos(k) / ((k * k) * t_m)) * ((l ** 2.0d0) / (k ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.3e-18) {
		tmp = 2.0 / ((((k * t_m) / l) * (((Math.sin(k) * t_m) / l) * t_m)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	} else {
		tmp = 2.0 * ((Math.cos(k) / ((k * k) * t_m)) * (Math.pow(l, 2.0) / Math.pow(k, 2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 2.3e-18:
		tmp = 2.0 / ((((k * t_m) / l) * (((math.sin(k) * t_m) / l) * t_m)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	else:
		tmp = 2.0 * ((math.cos(k) / ((k * k) * t_m)) * (math.pow(l, 2.0) / math.pow(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 (k <= 2.3e-18)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * t_m) / l) * Float64(Float64(Float64(sin(k) * t_m) / l) * t_m)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(Float64(k * k) * t_m)) * Float64((l ^ 2.0) / (k ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 2.3e-18)
		tmp = 2.0 / ((((k * t_m) / l) * (((sin(k) * t_m) / l) * t_m)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	else
		tmp = 2.0 * ((cos(k) / ((k * k) * t_m)) * ((l ^ 2.0) / (k ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.3e-18], N[(2.0 / N[(N[(N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $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[(N[Cos[k], $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.3000000000000001e-18

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. unpow3 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 67.6

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 67.6%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 74.5

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 74.5

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 74.5%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right) \cdot t\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right)} \cdot t\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)}\right) \cdot t\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \frac{\sin k \cdot t}{\ell}\right)} \cdot t\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\color{blue}{\frac{t}{\ell}} \cdot \tan k\right) \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot t\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot \tan k}{\ell}} \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot t\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot \tan k}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot \tan k}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{t \cdot \tan k}{\ell}} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\tan k \cdot t}}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\tan k \cdot t}}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 78.4

          \[\leadsto \frac{2}{\left(\frac{\tan k \cdot t}{\ell} \cdot \color{blue}{\left(\frac{\sin k \cdot t}{\ell} \cdot t\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Applied rewrites 78.4%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{\tan k \cdot t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{\color{blue}{k} \cdot t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 72.0%

       \[\leadsto \frac{2}{\left(\frac{\color{blue}{k} \cdot t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot t\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   if 2.3000000000000001e-18 < k

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 60.0

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 60.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. pow2 N/A

         \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto 2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      10. times-frac N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{{\sin k}^{2}}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{{\sin k}^{2}}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{\ell \cdot \color{blue}{\ell}}{{\sin k}^{2}}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{{k}^{2} \cdot t} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2}}\right) \]

      17. lower-/.f64 61.2

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\color{blue}{{\sin k}^{2}}}\right) \]

      18. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{\sin k}^{\color{blue}{2}}}\right) \]

      19. unpow2 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \color{blue}{\sin k}}\right) \]

      20. lift-sin.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \sin \color{blue}{k}}\right) \]

      21. lift-sin.f64 N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \sin k}\right) \]

      22. sqr-sin-a N/A

          \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot k\right)}}\right) \]

   6. Applied rewrites 56.9%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell \cdot \ell}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}\right) \]

   7. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2}}}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2}}{{k}^{\color{blue}{2}}}\right) \]

      2. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2}}{{k}^{2}}\right) \]

      3. lower-pow.f64 55.5

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2}}{{k}^{2}}\right) \]

   9. Applied rewrites 55.5%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 72.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 4.7 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot t\_m, 0.16666666666666666, t\_m\right)}{\ell} \cdot k\right) \cdot k\right) \cdot \left(\frac{t\_m}{\ell} \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)\right)\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(t\_m \cdot k\right) \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)} \cdot \ell\\ \end{array} \end{array} \]

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 (<= l 4.7e-18)
    (/
     2.0
     (*
      (*
       (* (* (/ (fma (* (* k k) t_m) 0.16666666666666666 t_m) l) k) k)
       (* (/ t_m l) (fma (/ k t_m) (/ k t_m) 2.0)))
      t_m))
    (* (/ l (* (* (* t_m k) t_m) (* k t_m))) l))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 4.7e-18) {
		tmp = 2.0 / (((((fma(((k * k) * t_m), 0.16666666666666666, t_m) / l) * k) * k) * ((t_m / l) * fma((k / t_m), (k / t_m), 2.0))) * t_m);
	} else {
		tmp = (l / (((t_m * k) * t_m) * (k * t_m))) * l;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 4.7e-18)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(fma(Float64(Float64(k * k) * t_m), 0.16666666666666666, t_m) / l) * k) * k) * Float64(Float64(t_m / l) * fma(Float64(k / t_m), Float64(k / t_m), 2.0))) * t_m));
	else
		tmp = Float64(Float64(l / Float64(Float64(Float64(t_m * k) * t_m) * Float64(k * t_m))) * 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[l, 4.7e-18], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * 0.16666666666666666 + t$95$m), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(N[(t$95$m * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4.6999999999999996e-18

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. unpow3 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 67.6

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 67.6%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 74.5

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 74.5

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 74.5%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot t}{\ell}\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell} + \frac{t}{\ell}\right)\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell} + \frac{t}{\ell}\right)}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \left(\color{blue}{\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell}} + \frac{t}{\ell}\right)\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{\frac{{k}^{2} \cdot t}{\ell}}, \frac{t}{\ell}\right)\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2} \cdot t}{\color{blue}{\ell}}, \frac{t}{\ell}\right)\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2} \cdot t}{\ell}, \frac{t}{\ell}\right)\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2} \cdot t}{\ell}, \frac{t}{\ell}\right)\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-/.f64 64.0

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(0.16666666666666666, \frac{{k}^{2} \cdot t}{\ell}, \frac{t}{\ell}\right)\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Applied rewrites 64.0%

      \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \mathsf{fma}\left(0.16666666666666666, \frac{{k}^{2} \cdot t}{\ell}, \frac{t}{\ell}\right)\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2} \cdot t}{\ell}, \frac{t}{\ell}\right)\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2} \cdot t}{\ell}, \frac{t}{\ell}\right)\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2} \cdot t}{\ell}, \frac{t}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2} \cdot t}{\ell}, \frac{t}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot t}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t}{\ell} \cdot \left({k}^{2} \cdot \mathsf{fma}\left(\frac{1}{6}, \frac{{k}^{2} \cdot t}{\ell}, \frac{t}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot t}} \]

   10. Applied rewrites 70.7%

       \[\leadsto \color{blue}{\frac{2}{\left(\left(\left(\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, 0.16666666666666666, t\right)}{\ell} \cdot k\right) \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)\right) \cdot t}} \]

   if 4.6999999999999996e-18 < l

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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 51.1

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.7

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      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 60.1

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 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. lift-*.f64 60.1

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 60.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   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 60.1

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      13. lower-*.f64 63.1

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 63.1%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lower-*.f64 66.6

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      7. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      8. lower-*.f64 66.6

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 66.6%

       \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 67.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.4 \cdot 10^{-19}:\\ \;\;\;\;\frac{\ell}{\left(\left(t\_m \cdot k\right) \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t\_m}\\ \end{array} \end{array} \]

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 6.4e-19)
    (* (/ l (* (* (* t_m k) t_m) (* k t_m))) l)
    (* 2.0 (/ (pow l 2.0) (* (pow k 4.0) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.4e-19) {
		tmp = (l / (((t_m * k) * t_m) * (k * t_m))) * l;
	} else {
		tmp = 2.0 * (pow(l, 2.0) / (pow(k, 4.0) * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6.4d-19) then
        tmp = (l / (((t_m * k) * t_m) * (k * t_m))) * l
    else
        tmp = 2.0d0 * ((l ** 2.0d0) / ((k ** 4.0d0) * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.4e-19) {
		tmp = (l / (((t_m * k) * t_m) * (k * t_m))) * l;
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) / (Math.pow(k, 4.0) * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 6.4e-19:
		tmp = (l / (((t_m * k) * t_m) * (k * t_m))) * l
	else:
		tmp = 2.0 * (math.pow(l, 2.0) / (math.pow(k, 4.0) * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6.4e-19)
		tmp = Float64(Float64(l / Float64(Float64(Float64(t_m * k) * t_m) * Float64(k * t_m))) * l);
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64((k ^ 4.0) * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 6.4e-19)
		tmp = (l / (((t_m * k) * t_m) * (k * t_m))) * l;
	else
		tmp = 2.0 * ((l ^ 2.0) / ((k ^ 4.0) * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6.4e-19], N[(N[(l / N[(N[(N[(t$95$m * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[k, 4.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.39999999999999965e-19

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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 51.1

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.7

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      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 60.1

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 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. lift-*.f64 60.1

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 60.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   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 60.1

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      13. lower-*.f64 63.1

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 63.1%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lower-*.f64 66.6

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      7. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      8. lower-*.f64 66.6

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 66.6%

       \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   if 6.39999999999999965e-19 < k

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 60.0

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 60.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

      2. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

      4. lower-pow.f64 51.8

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

   7. Applied rewrites 51.8%

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 67.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{+30}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t\_m}}{t\_m \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(t\_m \cdot k\right) \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)} \cdot \ell\\ \end{array} \end{array} \]

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 1e+30)
    (/ (/ (* l (/ l (* k k))) t_m) (* t_m t_m))
    (* (/ l (* (* (* t_m k) t_m) (* k t_m))) l))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1e+30) {
		tmp = ((l * (l / (k * k))) / t_m) / (t_m * t_m);
	} else {
		tmp = (l / (((t_m * k) * t_m) * (k * t_m))) * l;
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1d+30) then
        tmp = ((l * (l / (k * k))) / t_m) / (t_m * t_m)
    else
        tmp = (l / (((t_m * k) * t_m) * (k * t_m))) * l
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1e+30) {
		tmp = ((l * (l / (k * k))) / t_m) / (t_m * t_m);
	} else {
		tmp = (l / (((t_m * k) * t_m) * (k * t_m))) * l;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1e+30:
		tmp = ((l * (l / (k * k))) / t_m) / (t_m * t_m)
	else:
		tmp = (l / (((t_m * k) * t_m) * (k * t_m))) * l
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1e+30)
		tmp = Float64(Float64(Float64(l * Float64(l / Float64(k * k))) / t_m) / Float64(t_m * t_m));
	else
		tmp = Float64(Float64(l / Float64(Float64(Float64(t_m * k) * t_m) * Float64(k * t_m))) * l);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1e+30)
		tmp = ((l * (l / (k * k))) / t_m) / (t_m * t_m);
	else
		tmp = (l / (((t_m * k) * t_m) * (k * t_m))) * l;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1e+30], 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], N[(N[(l / N[(N[(N[(t$95$m * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1e30

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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 51.1

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 1e30 < t

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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 51.1

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.7

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      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 60.1

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 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. lift-*.f64 60.1

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 60.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   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 60.1

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      13. lower-*.f64 63.1

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 63.1%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lower-*.f64 66.6

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      7. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      8. lower-*.f64 66.6

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 66.6%

       \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 67.4% accurate, 5.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2 \cdot 10^{+145}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot k}}{\left(t\_m \cdot t\_m\right) \cdot k} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(t\_m \cdot k\right) \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)} \cdot \ell\\ \end{array} \end{array} \]

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 2e+145)
    (* (/ (/ l (* t_m k)) (* (* t_m t_m) k)) l)
    (* (/ l (* (* (* t_m k) t_m) (* k t_m))) l))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2e+145) {
		tmp = ((l / (t_m * k)) / ((t_m * t_m) * k)) * l;
	} else {
		tmp = (l / (((t_m * k) * t_m) * (k * t_m))) * l;
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2d+145) then
        tmp = ((l / (t_m * k)) / ((t_m * t_m) * k)) * l
    else
        tmp = (l / (((t_m * k) * t_m) * (k * t_m))) * l
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2e+145) {
		tmp = ((l / (t_m * k)) / ((t_m * t_m) * k)) * l;
	} else {
		tmp = (l / (((t_m * k) * t_m) * (k * t_m))) * l;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2e+145:
		tmp = ((l / (t_m * k)) / ((t_m * t_m) * k)) * l
	else:
		tmp = (l / (((t_m * k) * t_m) * (k * t_m))) * l
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2e+145)
		tmp = Float64(Float64(Float64(l / Float64(t_m * k)) / Float64(Float64(t_m * t_m) * k)) * l);
	else
		tmp = Float64(Float64(l / Float64(Float64(Float64(t_m * k) * t_m) * Float64(k * t_m))) * l);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2e+145)
		tmp = ((l / (t_m * k)) / ((t_m * t_m) * k)) * l;
	else
		tmp = (l / (((t_m * k) * t_m) * (k * t_m))) * l;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2e+145], N[(N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(l / N[(N[(N[(t$95$m * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2e145

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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 51.1

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.7

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      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 60.1

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 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. lift-*.f64 60.1

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 60.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   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 60.1

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      13. lower-*.f64 63.1

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 63.1%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      4. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell}{k \cdot t}}{k \cdot \left(t \cdot t\right)} \cdot \ell \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k \cdot t}}{k \cdot \left(t \cdot t\right)} \cdot \ell \]

      6. lower-/.f64 64.6

         \[\leadsto \frac{\frac{\ell}{k \cdot t}}{k \cdot \left(t \cdot t\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k \cdot t}}{k \cdot \left(t \cdot t\right)} \cdot \ell \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\ell}{t \cdot k}}{k \cdot \left(t \cdot t\right)} \cdot \ell \]

      9. lower-*.f64 64.6

         \[\leadsto \frac{\frac{\ell}{t \cdot k}}{k \cdot \left(t \cdot t\right)} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t \cdot k}}{k \cdot \left(t \cdot t\right)} \cdot \ell \]

      11. *-commutative N/A

          \[\leadsto \frac{\frac{\ell}{t \cdot k}}{\left(t \cdot t\right) \cdot k} \cdot \ell \]

      12. lower-*.f64 64.6

          \[\leadsto \frac{\frac{\ell}{t \cdot k}}{\left(t \cdot t\right) \cdot k} \cdot \ell \]

   10. Applied rewrites 64.6%

       \[\leadsto \frac{\frac{\ell}{t \cdot k}}{\left(t \cdot t\right) \cdot k} \cdot \ell \]

   if 2e145 < t

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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 51.1

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.7

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      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 60.1

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 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. lift-*.f64 60.1

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 60.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   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 60.1

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      13. lower-*.f64 63.1

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 63.1%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lower-*.f64 66.6

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      7. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      8. lower-*.f64 66.6

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 66.6%

       \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 66.6% accurate, 5.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 7.8 \cdot 10^{-144}:\\ \;\;\;\;\frac{\ell}{\left(k \cdot \left(t\_m \cdot t\_m\right)\right) \cdot t\_m} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(t\_m \cdot k\right) \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)} \cdot \ell\\ \end{array} \end{array} \]

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 (<= l 7.8e-144)
    (* (/ l (* (* k (* t_m t_m)) t_m)) (/ l k))
    (* (/ l (* (* (* t_m k) t_m) (* k t_m))) l))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 7.8e-144) {
		tmp = (l / ((k * (t_m * t_m)) * t_m)) * (l / k);
	} else {
		tmp = (l / (((t_m * k) * t_m) * (k * t_m))) * l;
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 7.8d-144) then
        tmp = (l / ((k * (t_m * t_m)) * t_m)) * (l / k)
    else
        tmp = (l / (((t_m * k) * t_m) * (k * t_m))) * l
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 7.8e-144) {
		tmp = (l / ((k * (t_m * t_m)) * t_m)) * (l / k);
	} else {
		tmp = (l / (((t_m * k) * t_m) * (k * t_m))) * l;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 7.8e-144:
		tmp = (l / ((k * (t_m * t_m)) * t_m)) * (l / k)
	else:
		tmp = (l / (((t_m * k) * t_m) * (k * t_m))) * l
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 7.8e-144)
		tmp = Float64(Float64(l / Float64(Float64(k * Float64(t_m * t_m)) * t_m)) * Float64(l / k));
	else
		tmp = Float64(Float64(l / Float64(Float64(Float64(t_m * k) * t_m) * Float64(k * t_m))) * l);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 7.8e-144)
		tmp = (l / ((k * (t_m * t_m)) * t_m)) * (l / k);
	else
		tmp = (l / (((t_m * k) * t_m) * (k * t_m))) * l;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 7.8e-144], N[(N[(l / N[(N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(N[(t$95$m * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 7.8000000000000003e-144

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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 51.1

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.7

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      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 60.1

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 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. lift-*.f64 60.1

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 60.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   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. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      5. times-frac N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \frac{\color{blue}{\ell}}{k} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{k} \]

      11. pow3 N/A

          \[\leadsto \frac{\ell}{{t}^{3} \cdot k} \cdot \frac{\ell}{k} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\ell}{{t}^{3} \cdot k} \cdot \frac{\ell}{k} \]

      13. *-commutative N/A

          \[\leadsto \frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{k} \]

      15. pow3 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \frac{\ell}{k} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \frac{\ell}{k} \]

      17. associate-*r* N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot t} \cdot \frac{\ell}{k} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot t} \cdot \frac{\ell}{k} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot t} \cdot \frac{\ell}{k} \]

      20. lower-/.f64 63.9

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot t} \cdot \frac{\ell}{\color{blue}{k}} \]

   8. Applied rewrites 63.9%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot t} \cdot \color{blue}{\frac{\ell}{k}} \]

   if 7.8000000000000003e-144 < l

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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 51.1

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.7

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      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 60.1

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 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. lift-*.f64 60.1

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 60.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   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 60.1

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      13. lower-*.f64 63.1

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 63.1%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lower-*.f64 66.6

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      7. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      8. lower-*.f64 66.6

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 66.6%

       \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 65.5% accurate, 6.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\ell}{\left(\left(t\_m \cdot k\right) \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)} \cdot \ell\right) \end{array} \]

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 k) t_m) (* k t_m))) l)))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / (((t_m * k) * t_m) * (k * t_m))) * 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 * k) * t_m) * (k * t_m))) * 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 * k) * t_m) * (k * t_m))) * 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 * k) * t_m) * (k * t_m))) * 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(Float64(t_m * k) * t_m) * Float64(k * t_m))) * 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 * k) * t_m) * (k * t_m))) * 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[(N[(t$95$m * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. 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 51.1

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 51.1%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
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 55.7

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   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 60.1

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   14. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   15. pow3 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. lift-*.f64 60.1

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

6. Applied rewrites 60.1%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
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 60.1

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   5. *-commutative N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   8. associate-*l* N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

   9. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   10. *-commutative N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   13. lower-*.f64 63.1

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

8. Applied rewrites 63.1%

   \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   3. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   5. lower-*.f64 66.6

      \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   7. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. lower-*.f64 66.6

      \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

10. Applied rewrites 66.6%

    \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
11. Add Preprocessing

Alternative 18: 63.1% accurate, 6.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\ell}{\left(k \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \left(k \cdot t\_m\right)} \cdot \ell\right) \end{array} \]

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 (* (* k (* t_m t_m)) (* k t_m))) l)))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / ((k * (t_m * t_m)) * (k * t_m))) * 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 / ((k * (t_m * t_m)) * (k * t_m))) * 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 / ((k * (t_m * t_m)) * (k * t_m))) * 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 / ((k * (t_m * t_m)) * (k * t_m))) * 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(k * Float64(t_m * t_m)) * Float64(k * t_m))) * 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 / ((k * (t_m * t_m)) * (k * t_m))) * 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[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. 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 51.1

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 51.1%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
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 55.7

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   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 60.1

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   14. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   15. pow3 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. lift-*.f64 60.1

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

6. Applied rewrites 60.1%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
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 60.1

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   5. *-commutative N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   8. associate-*l* N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

   9. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   10. *-commutative N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   13. lower-*.f64 63.1

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

8. Applied rewrites 63.1%

   \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025162 
(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))))