Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.6% → 84.3%

Time: 9.4s

Alternatives: 29

Speedup: 1.6×

• 27.1% 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: 54.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \log l\_m \cdot 2\\ t_3 := \log t\_m \cdot 3\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\frac{t\_3 \cdot t\_3 - t\_2 \cdot t\_2}{t\_3 + t\_2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right) + 1\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (* (log l_m) 2.0)) (t_3 (* (log t_m) 3.0)))
   (*
    t_s
    (if (<= t_m 9e-6)
      (/ 2.0 (* (* (/ k l_m) (/ k l_m)) (/ (* t_m (pow (sin k) 2.0)) (cos k))))
      (/
       2.0
       (*
        (*
         (* (exp (/ (- (* t_3 t_3) (* t_2 t_2)) (+ t_3 t_2))) (sin k))
         (tan k))
        (+ (+ 1.0 (* (/ k t_m) (/ k t_m))) 1.0)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = log(l_m) * 2.0;
	double t_3 = log(t_m) * 3.0;
	double tmp;
	if (t_m <= 9e-6) {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * pow(sin(k), 2.0)) / cos(k)));
	} else {
		tmp = 2.0 / (((exp((((t_3 * t_3) - (t_2 * t_2)) / (t_3 + t_2))) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = log(l_m) * 2.0d0
    t_3 = log(t_m) * 3.0d0
    if (t_m <= 9d-6) then
        tmp = 2.0d0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ** 2.0d0)) / cos(k)))
    else
        tmp = 2.0d0 / (((exp((((t_3 * t_3) - (t_2 * t_2)) / (t_3 + t_2))) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) * (k / t_m))) + 1.0d0))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = Math.log(l_m) * 2.0;
	double t_3 = Math.log(t_m) * 3.0;
	double tmp;
	if (t_m <= 9e-6) {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(k)));
	} else {
		tmp = 2.0 / (((Math.exp((((t_3 * t_3) - (t_2 * t_2)) / (t_3 + t_2))) * Math.sin(k)) * Math.tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	t_2 = math.log(l_m) * 2.0
	t_3 = math.log(t_m) * 3.0
	tmp = 0
	if t_m <= 9e-6:
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * math.pow(math.sin(k), 2.0)) / math.cos(k)))
	else:
		tmp = 2.0 / (((math.exp((((t_3 * t_3) - (t_2 * t_2)) / (t_3 + t_2))) * math.sin(k)) * math.tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(log(l_m) * 2.0)
	t_3 = Float64(log(t_m) * 3.0)
	tmp = 0.0
	if (t_m <= 9e-6)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(Float64(t_3 * t_3) - Float64(t_2 * t_2)) / Float64(t_3 + t_2))) * sin(k)) * tan(k)) * Float64(Float64(1.0 + Float64(Float64(k / t_m) * Float64(k / t_m))) + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	t_2 = log(l_m) * 2.0;
	t_3 = log(t_m) * 3.0;
	tmp = 0.0;
	if (t_m <= 9e-6)
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ^ 2.0)) / cos(k)));
	else
		tmp = 2.0 / (((exp((((t_3 * t_3) - (t_2 * t_2)) / (t_3 + t_2))) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := Block[{t$95$2 = N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9e-6], N[(2.0 / N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 9.00000000000000023e-6

   1. Initial program 48.7%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 55.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 55.0

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

   8. Applied rewrites 55.0%

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

   9. Taylor expanded in t around 0

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

       1. times-frac N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. pow2 N/A

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

       5. times-frac N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-pow.f64 N/A

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

       12. lift-sin.f64 N/A

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

       13. lower-cos.f64 72.8

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

   11. Applied rewrites 72.8%

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

   if 9.00000000000000023e-6 < t

   1. Initial program 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 42.3

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

   4. Applied rewrites 42.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 42.3

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

   6. Applied rewrites 42.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

      8. flip-- N/A

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

      9. lower-/.f64 N/A

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

   8. Applied rewrites 42.4%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\frac{\left(\log t \cdot 3\right) \cdot \left(\log t \cdot 3\right) - \left(\log \ell \cdot 2\right) \cdot \left(\log \ell \cdot 2\right)}{\log t \cdot 3 + \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-184}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{{t\_m}^{3}}{l\_m}}{l\_m} \cdot \left(\sin k \cdot \tan k\right)}}{\left(\frac{k \cdot k}{t\_m \cdot t\_m} + 1\right) + 1}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\_m\right)}^{2} \cdot \left(\left(k \cdot k\right) \cdot 0.3333333333333333 + 2\right) + {k}^{4}}{l\_m \cdot l\_m} \cdot t\_m}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+275}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2
         (/
          2.0
          (*
           (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k))
           (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))))
   (*
    t_s
    (if (<= t_2 -5e-184)
      (/
       (/ 2.0 (* (/ (/ (pow t_m 3.0) l_m) l_m) (* (sin k) (tan k))))
       (+ (+ (/ (* k k) (* t_m t_m)) 1.0) 1.0))
      (if (<= t_2 0.0)
        (/
         2.0
         (*
          (/
           (+
            (* (pow (* k t_m) 2.0) (+ (* (* k k) 0.3333333333333333) 2.0))
            (pow k 4.0))
           (* l_m l_m))
          t_m))
        (if (<= t_2 2e+275)
          t_2
          (/
           2.0
           (*
            (* (/ k l_m) (/ k l_m))
            (/ (* t_m (pow (sin k) 2.0)) (cos k))))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = 2.0 / ((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	double tmp;
	if (t_2 <= -5e-184) {
		tmp = (2.0 / (((pow(t_m, 3.0) / l_m) / l_m) * (sin(k) * tan(k)))) / ((((k * k) / (t_m * t_m)) + 1.0) + 1.0);
	} else if (t_2 <= 0.0) {
		tmp = 2.0 / ((((pow((k * t_m), 2.0) * (((k * k) * 0.3333333333333333) + 2.0)) + pow(k, 4.0)) / (l_m * l_m)) * t_m);
	} else if (t_2 <= 2e+275) {
		tmp = t_2;
	} else {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * pow(sin(k), 2.0)) / cos(k)));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 2.0d0 / (((((t_m ** 3.0d0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    if (t_2 <= (-5d-184)) then
        tmp = (2.0d0 / ((((t_m ** 3.0d0) / l_m) / l_m) * (sin(k) * tan(k)))) / ((((k * k) / (t_m * t_m)) + 1.0d0) + 1.0d0)
    else if (t_2 <= 0.0d0) then
        tmp = 2.0d0 / ((((((k * t_m) ** 2.0d0) * (((k * k) * 0.3333333333333333d0) + 2.0d0)) + (k ** 4.0d0)) / (l_m * l_m)) * t_m)
    else if (t_2 <= 2d+275) then
        tmp = t_2
    else
        tmp = 2.0d0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ** 2.0d0)) / cos(k)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = 2.0 / ((((Math.pow(t_m, 3.0) / (l_m * l_m)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	double tmp;
	if (t_2 <= -5e-184) {
		tmp = (2.0 / (((Math.pow(t_m, 3.0) / l_m) / l_m) * (Math.sin(k) * Math.tan(k)))) / ((((k * k) / (t_m * t_m)) + 1.0) + 1.0);
	} else if (t_2 <= 0.0) {
		tmp = 2.0 / ((((Math.pow((k * t_m), 2.0) * (((k * k) * 0.3333333333333333) + 2.0)) + Math.pow(k, 4.0)) / (l_m * l_m)) * t_m);
	} else if (t_2 <= 2e+275) {
		tmp = t_2;
	} else {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(k)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	t_2 = 2.0 / ((((math.pow(t_m, 3.0) / (l_m * l_m)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	tmp = 0
	if t_2 <= -5e-184:
		tmp = (2.0 / (((math.pow(t_m, 3.0) / l_m) / l_m) * (math.sin(k) * math.tan(k)))) / ((((k * k) / (t_m * t_m)) + 1.0) + 1.0)
	elif t_2 <= 0.0:
		tmp = 2.0 / ((((math.pow((k * t_m), 2.0) * (((k * k) * 0.3333333333333333) + 2.0)) + math.pow(k, 4.0)) / (l_m * l_m)) * t_m)
	elif t_2 <= 2e+275:
		tmp = t_2
	else:
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * math.pow(math.sin(k), 2.0)) / math.cos(k)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)))
	tmp = 0.0
	if (t_2 <= -5e-184)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64((t_m ^ 3.0) / l_m) / l_m) * Float64(sin(k) * tan(k)))) / Float64(Float64(Float64(Float64(k * k) / Float64(t_m * t_m)) + 1.0) + 1.0));
	elseif (t_2 <= 0.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * Float64(Float64(Float64(k * k) * 0.3333333333333333) + 2.0)) + (k ^ 4.0)) / Float64(l_m * l_m)) * t_m));
	elseif (t_2 <= 2e+275)
		tmp = t_2;
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	t_2 = 2.0 / (((((t_m ^ 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	tmp = 0.0;
	if (t_2 <= -5e-184)
		tmp = (2.0 / ((((t_m ^ 3.0) / l_m) / l_m) * (sin(k) * tan(k)))) / ((((k * k) / (t_m * t_m)) + 1.0) + 1.0);
	elseif (t_2 <= 0.0)
		tmp = 2.0 / ((((((k * t_m) ^ 2.0) * (((k * k) * 0.3333333333333333) + 2.0)) + (k ^ 4.0)) / (l_m * l_m)) * t_m);
	elseif (t_2 <= 2e+275)
		tmp = t_2;
	else
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ^ 2.0)) / cos(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := Block[{t$95$2 = N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -5e-184], N[(N[(2.0 / N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(k * k), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+275], t$95$2, N[(2.0 / N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < -5.00000000000000003e-184

   1. Initial program 83.4%

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

   4. Applied rewrites 79.5%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 63.1

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

   6. Applied rewrites 63.1%

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

   if -5.00000000000000003e-184 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 0.0

   1. Initial program 77.8%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 69.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(2 + \frac{1}{3} \cdot {k}^{2}\right)\right)}{{\ell}^{2}} + \frac{{k}^{4}}{{\ell}^{2}}\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(2 + \frac{1}{3} \cdot {k}^{2}\right)\right)}{{\ell}^{2}} + \frac{{k}^{4}}{{\ell}^{2}}\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(2 + \frac{1}{3} \cdot {k}^{2}\right)\right)}{{\ell}^{2}} + \frac{{k}^{4}}{{\ell}^{2}}\right) \cdot t} \]

   8. Applied rewrites 83.6%

      \[\leadsto \frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot \left(\left(k \cdot k\right) \cdot 0.3333333333333333 + 2\right) + {k}^{4}}{\ell \cdot \ell} \cdot \color{blue}{t}} \]

   if 0.0 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 1.99999999999999992e275

   1. Initial program 95.8%

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

   if 1.99999999999999992e275 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 18.5%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 40.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.0

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

   8. Applied rewrites 45.0%

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

   9. Taylor expanded in t around 0

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

       1. times-frac N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. pow2 N/A

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

       5. times-frac N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-pow.f64 N/A

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

       12. lift-sin.f64 N/A

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

       13. lower-cos.f64 77.0

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

   11. Applied rewrites 77.0%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq -5 \cdot 10^{-184}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\left(\frac{k \cdot k}{t \cdot t} + 1\right) + 1}\\ \mathbf{elif}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 0:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot \left(\left(k \cdot k\right) \cdot 0.3333333333333333 + 2\right) + {k}^{4}}{\ell \cdot \ell} \cdot t}\\ \mathbf{elif}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{k \cdot k}{t\_m \cdot t\_m}\\ t_3 := \frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-184}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{{t\_m}^{3}}{l\_m}}{l\_m} \cdot \left(\sin k \cdot \tan k\right)}}{\left(t\_2 + 1\right) + 1}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\_m\right)}^{2} \cdot \left(\left(k \cdot k\right) \cdot 0.3333333333333333 + 2\right) + {k}^{4}}{l\_m \cdot l\_m} \cdot t\_m}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + t\_2\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (/ (* k k) (* t_m t_m)))
        (t_3
         (/
          2.0
          (*
           (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k))
           (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))))
   (*
    t_s
    (if (<= t_3 -5e-184)
      (/
       (/ 2.0 (* (/ (/ (pow t_m 3.0) l_m) l_m) (* (sin k) (tan k))))
       (+ (+ t_2 1.0) 1.0))
      (if (<= t_3 0.0)
        (/
         2.0
         (*
          (/
           (+
            (* (pow (* k t_m) 2.0) (+ (* (* k k) 0.3333333333333333) 2.0))
            (pow k 4.0))
           (* l_m l_m))
          t_m))
        (if (<= t_3 2e+275)
          (/
           2.0
           (*
            (* (* (/ (* (* t_m t_m) t_m) (* l_m l_m)) (sin k)) (tan k))
            (+ (+ 1.0 t_2) 1.0)))
          (/
           2.0
           (*
            (* (/ k l_m) (/ k l_m))
            (/ (* t_m (pow (sin k) 2.0)) (cos k))))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = (k * k) / (t_m * t_m);
	double t_3 = 2.0 / ((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	double tmp;
	if (t_3 <= -5e-184) {
		tmp = (2.0 / (((pow(t_m, 3.0) / l_m) / l_m) * (sin(k) * tan(k)))) / ((t_2 + 1.0) + 1.0);
	} else if (t_3 <= 0.0) {
		tmp = 2.0 / ((((pow((k * t_m), 2.0) * (((k * k) * 0.3333333333333333) + 2.0)) + pow(k, 4.0)) / (l_m * l_m)) * t_m);
	} else if (t_3 <= 2e+275) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + t_2) + 1.0));
	} else {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * pow(sin(k), 2.0)) / cos(k)));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = (k * k) / (t_m * t_m)
    t_3 = 2.0d0 / (((((t_m ** 3.0d0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    if (t_3 <= (-5d-184)) then
        tmp = (2.0d0 / ((((t_m ** 3.0d0) / l_m) / l_m) * (sin(k) * tan(k)))) / ((t_2 + 1.0d0) + 1.0d0)
    else if (t_3 <= 0.0d0) then
        tmp = 2.0d0 / ((((((k * t_m) ** 2.0d0) * (((k * k) * 0.3333333333333333d0) + 2.0d0)) + (k ** 4.0d0)) / (l_m * l_m)) * t_m)
    else if (t_3 <= 2d+275) then
        tmp = 2.0d0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0d0 + t_2) + 1.0d0))
    else
        tmp = 2.0d0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ** 2.0d0)) / cos(k)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = (k * k) / (t_m * t_m);
	double t_3 = 2.0 / ((((Math.pow(t_m, 3.0) / (l_m * l_m)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	double tmp;
	if (t_3 <= -5e-184) {
		tmp = (2.0 / (((Math.pow(t_m, 3.0) / l_m) / l_m) * (Math.sin(k) * Math.tan(k)))) / ((t_2 + 1.0) + 1.0);
	} else if (t_3 <= 0.0) {
		tmp = 2.0 / ((((Math.pow((k * t_m), 2.0) * (((k * k) * 0.3333333333333333) + 2.0)) + Math.pow(k, 4.0)) / (l_m * l_m)) * t_m);
	} else if (t_3 <= 2e+275) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + t_2) + 1.0));
	} else {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(k)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	t_2 = (k * k) / (t_m * t_m)
	t_3 = 2.0 / ((((math.pow(t_m, 3.0) / (l_m * l_m)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	tmp = 0
	if t_3 <= -5e-184:
		tmp = (2.0 / (((math.pow(t_m, 3.0) / l_m) / l_m) * (math.sin(k) * math.tan(k)))) / ((t_2 + 1.0) + 1.0)
	elif t_3 <= 0.0:
		tmp = 2.0 / ((((math.pow((k * t_m), 2.0) * (((k * k) * 0.3333333333333333) + 2.0)) + math.pow(k, 4.0)) / (l_m * l_m)) * t_m)
	elif t_3 <= 2e+275:
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * math.sin(k)) * math.tan(k)) * ((1.0 + t_2) + 1.0))
	else:
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * math.pow(math.sin(k), 2.0)) / math.cos(k)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(Float64(k * k) / Float64(t_m * t_m))
	t_3 = Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)))
	tmp = 0.0
	if (t_3 <= -5e-184)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64((t_m ^ 3.0) / l_m) / l_m) * Float64(sin(k) * tan(k)))) / Float64(Float64(t_2 + 1.0) + 1.0));
	elseif (t_3 <= 0.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * Float64(Float64(Float64(k * k) * 0.3333333333333333) + 2.0)) + (k ^ 4.0)) / Float64(l_m * l_m)) * t_m));
	elseif (t_3 <= 2e+275)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + t_2) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	t_2 = (k * k) / (t_m * t_m);
	t_3 = 2.0 / (((((t_m ^ 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	tmp = 0.0;
	if (t_3 <= -5e-184)
		tmp = (2.0 / ((((t_m ^ 3.0) / l_m) / l_m) * (sin(k) * tan(k)))) / ((t_2 + 1.0) + 1.0);
	elseif (t_3 <= 0.0)
		tmp = 2.0 / ((((((k * t_m) ^ 2.0) * (((k * k) * 0.3333333333333333) + 2.0)) + (k ^ 4.0)) / (l_m * l_m)) * t_m);
	elseif (t_3 <= 2e+275)
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + t_2) + 1.0));
	else
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ^ 2.0)) / cos(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := Block[{t$95$2 = N[(N[(k * k), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, -5e-184], N[(N[(2.0 / N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$2 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+275], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < -5.00000000000000003e-184

   1. Initial program 83.4%

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

   4. Applied rewrites 79.5%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 63.1

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

   6. Applied rewrites 63.1%

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

   if -5.00000000000000003e-184 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 0.0

   1. Initial program 77.8%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 69.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(2 + \frac{1}{3} \cdot {k}^{2}\right)\right)}{{\ell}^{2}} + \frac{{k}^{4}}{{\ell}^{2}}\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(2 + \frac{1}{3} \cdot {k}^{2}\right)\right)}{{\ell}^{2}} + \frac{{k}^{4}}{{\ell}^{2}}\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(2 + \frac{1}{3} \cdot {k}^{2}\right)\right)}{{\ell}^{2}} + \frac{{k}^{4}}{{\ell}^{2}}\right) \cdot t} \]

   8. Applied rewrites 83.6%

      \[\leadsto \frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot \left(\left(k \cdot k\right) \cdot 0.3333333333333333 + 2\right) + {k}^{4}}{\ell \cdot \ell} \cdot \color{blue}{t}} \]

   if 0.0 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 1.99999999999999992e275

   1. Initial program 95.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. frac-times N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 95.5

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

   4. Applied rewrites 95.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 95.5

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

   6. Applied rewrites 95.5%

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

   if 1.99999999999999992e275 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 18.5%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 40.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.0

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

   8. Applied rewrites 45.0%

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

   9. Taylor expanded in t around 0

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

       1. times-frac N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. pow2 N/A

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

       5. times-frac N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-pow.f64 N/A

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

       12. lift-sin.f64 N/A

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

       13. lower-cos.f64 77.0

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

   11. Applied rewrites 77.0%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq -5 \cdot 10^{-184}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\left(\frac{k \cdot k}{t \cdot t} + 1\right) + 1}\\ \mathbf{elif}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 0:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot \left(\left(k \cdot k\right) \cdot 0.3333333333333333 + 2\right) + {k}^{4}}{\ell \cdot \ell} \cdot t}\\ \mathbf{elif}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k \cdot k}{t \cdot t}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-184}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t\_m}^{3}}{l\_m} \cdot \left(\tan k \cdot \sin k\right)}{l\_m} \cdot \left(\left(1 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right) + 1\right)}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\_m\right)}^{2} \cdot \left(\left(k \cdot k\right) \cdot 0.3333333333333333 + 2\right) + {k}^{4}}{l\_m \cdot l\_m} \cdot t\_m}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k \cdot k}{t\_m \cdot t\_m}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2
         (/
          2.0
          (*
           (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k))
           (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))))
   (*
    t_s
    (if (<= t_2 -5e-184)
      (/
       2.0
       (*
        (/ (* (/ (pow t_m 3.0) l_m) (* (tan k) (sin k))) l_m)
        (+ (+ 1.0 (* (/ k t_m) (/ k t_m))) 1.0)))
      (if (<= t_2 0.0)
        (/
         2.0
         (*
          (/
           (+
            (* (pow (* k t_m) 2.0) (+ (* (* k k) 0.3333333333333333) 2.0))
            (pow k 4.0))
           (* l_m l_m))
          t_m))
        (if (<= t_2 2e+275)
          (/
           2.0
           (*
            (* (* (/ (* (* t_m t_m) t_m) (* l_m l_m)) (sin k)) (tan k))
            (+ (+ 1.0 (/ (* k k) (* t_m t_m))) 1.0)))
          (/
           2.0
           (*
            (* (/ k l_m) (/ k l_m))
            (/ (* t_m (pow (sin k) 2.0)) (cos k))))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = 2.0 / ((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	double tmp;
	if (t_2 <= -5e-184) {
		tmp = 2.0 / ((((pow(t_m, 3.0) / l_m) * (tan(k) * sin(k))) / l_m) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	} else if (t_2 <= 0.0) {
		tmp = 2.0 / ((((pow((k * t_m), 2.0) * (((k * k) * 0.3333333333333333) + 2.0)) + pow(k, 4.0)) / (l_m * l_m)) * t_m);
	} else if (t_2 <= 2e+275) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + ((k * k) / (t_m * t_m))) + 1.0));
	} else {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * pow(sin(k), 2.0)) / cos(k)));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 2.0d0 / (((((t_m ** 3.0d0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    if (t_2 <= (-5d-184)) then
        tmp = 2.0d0 / (((((t_m ** 3.0d0) / l_m) * (tan(k) * sin(k))) / l_m) * ((1.0d0 + ((k / t_m) * (k / t_m))) + 1.0d0))
    else if (t_2 <= 0.0d0) then
        tmp = 2.0d0 / ((((((k * t_m) ** 2.0d0) * (((k * k) * 0.3333333333333333d0) + 2.0d0)) + (k ** 4.0d0)) / (l_m * l_m)) * t_m)
    else if (t_2 <= 2d+275) then
        tmp = 2.0d0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0d0 + ((k * k) / (t_m * t_m))) + 1.0d0))
    else
        tmp = 2.0d0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ** 2.0d0)) / cos(k)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = 2.0 / ((((Math.pow(t_m, 3.0) / (l_m * l_m)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	double tmp;
	if (t_2 <= -5e-184) {
		tmp = 2.0 / ((((Math.pow(t_m, 3.0) / l_m) * (Math.tan(k) * Math.sin(k))) / l_m) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	} else if (t_2 <= 0.0) {
		tmp = 2.0 / ((((Math.pow((k * t_m), 2.0) * (((k * k) * 0.3333333333333333) + 2.0)) + Math.pow(k, 4.0)) / (l_m * l_m)) * t_m);
	} else if (t_2 <= 2e+275) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + ((k * k) / (t_m * t_m))) + 1.0));
	} else {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(k)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	t_2 = 2.0 / ((((math.pow(t_m, 3.0) / (l_m * l_m)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	tmp = 0
	if t_2 <= -5e-184:
		tmp = 2.0 / ((((math.pow(t_m, 3.0) / l_m) * (math.tan(k) * math.sin(k))) / l_m) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0))
	elif t_2 <= 0.0:
		tmp = 2.0 / ((((math.pow((k * t_m), 2.0) * (((k * k) * 0.3333333333333333) + 2.0)) + math.pow(k, 4.0)) / (l_m * l_m)) * t_m)
	elif t_2 <= 2e+275:
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * math.sin(k)) * math.tan(k)) * ((1.0 + ((k * k) / (t_m * t_m))) + 1.0))
	else:
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * math.pow(math.sin(k), 2.0)) / math.cos(k)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)))
	tmp = 0.0
	if (t_2 <= -5e-184)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / l_m) * Float64(tan(k) * sin(k))) / l_m) * Float64(Float64(1.0 + Float64(Float64(k / t_m) * Float64(k / t_m))) + 1.0)));
	elseif (t_2 <= 0.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * Float64(Float64(Float64(k * k) * 0.3333333333333333) + 2.0)) + (k ^ 4.0)) / Float64(l_m * l_m)) * t_m));
	elseif (t_2 <= 2e+275)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + Float64(Float64(k * k) / Float64(t_m * t_m))) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	t_2 = 2.0 / (((((t_m ^ 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	tmp = 0.0;
	if (t_2 <= -5e-184)
		tmp = 2.0 / (((((t_m ^ 3.0) / l_m) * (tan(k) * sin(k))) / l_m) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	elseif (t_2 <= 0.0)
		tmp = 2.0 / ((((((k * t_m) ^ 2.0) * (((k * k) * 0.3333333333333333) + 2.0)) + (k ^ 4.0)) / (l_m * l_m)) * t_m);
	elseif (t_2 <= 2e+275)
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + ((k * k) / (t_m * t_m))) + 1.0));
	else
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ^ 2.0)) / cos(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := Block[{t$95$2 = N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -5e-184], N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+275], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(k * k), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < -5.00000000000000003e-184

   1. Initial program 83.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 7.6

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

   4. Applied rewrites 7.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 7.6

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

   6. Applied rewrites 7.6%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. exp-diff N/A

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

      13. pow-to-exp N/A

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

      14. pow-to-exp N/A

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

      15. pow2 N/A

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

      16. associate-/r* N/A

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

   8. Applied rewrites 79.3%

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

   if -5.00000000000000003e-184 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 0.0

   1. Initial program 77.8%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 69.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(2 + \frac{1}{3} \cdot {k}^{2}\right)\right)}{{\ell}^{2}} + \frac{{k}^{4}}{{\ell}^{2}}\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(2 + \frac{1}{3} \cdot {k}^{2}\right)\right)}{{\ell}^{2}} + \frac{{k}^{4}}{{\ell}^{2}}\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(2 + \frac{1}{3} \cdot {k}^{2}\right)\right)}{{\ell}^{2}} + \frac{{k}^{4}}{{\ell}^{2}}\right) \cdot t} \]

   8. Applied rewrites 83.6%

      \[\leadsto \frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot \left(\left(k \cdot k\right) \cdot 0.3333333333333333 + 2\right) + {k}^{4}}{\ell \cdot \ell} \cdot \color{blue}{t}} \]

   if 0.0 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 1.99999999999999992e275

   1. Initial program 95.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. frac-times N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 95.5

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

   4. Applied rewrites 95.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 95.5

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

   6. Applied rewrites 95.5%

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

   if 1.99999999999999992e275 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 18.5%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 40.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.0

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

   8. Applied rewrites 45.0%

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

   9. Taylor expanded in t around 0

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

       1. times-frac N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. pow2 N/A

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

       5. times-frac N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-pow.f64 N/A

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

       12. lift-sin.f64 N/A

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

       13. lower-cos.f64 77.0

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

   11. Applied rewrites 77.0%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq -5 \cdot 10^{-184}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\tan k \cdot \sin k\right)}{\ell} \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)}\\ \mathbf{elif}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 0:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot \left(\left(k \cdot k\right) \cdot 0.3333333333333333 + 2\right) + {k}^{4}}{\ell \cdot \ell} \cdot t}\\ \mathbf{elif}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k \cdot k}{t \cdot t}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-253}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{{\left(k \cdot t\_m\right)}^{2} \cdot t\_m}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot t\_m\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2
         (/
          2.0
          (*
           (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k))
           (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))))
   (*
    t_s
    (if (<= t_2 2e-253)
      (/ (* l_m l_m) (* (pow (* k t_m) 2.0) t_m))
      (if (<= t_2 2e+275)
        (/
         2.0
         (* (* (* (/ (* (* t_m t_m) t_m) (* l_m l_m)) (sin k)) (tan k)) 2.0))
        (/ 2.0 (* (* (* (/ k l_m) (/ k l_m)) t_m) (* k k))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = 2.0 / ((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	double tmp;
	if (t_2 <= 2e-253) {
		tmp = (l_m * l_m) / (pow((k * t_m), 2.0) * t_m);
	} else if (t_2 <= 2e+275) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * sin(k)) * tan(k)) * 2.0);
	} else {
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 2.0d0 / (((((t_m ** 3.0d0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    if (t_2 <= 2d-253) then
        tmp = (l_m * l_m) / (((k * t_m) ** 2.0d0) * t_m)
    else if (t_2 <= 2d+275) then
        tmp = 2.0d0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * sin(k)) * tan(k)) * 2.0d0)
    else
        tmp = 2.0d0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = 2.0 / ((((Math.pow(t_m, 3.0) / (l_m * l_m)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	double tmp;
	if (t_2 <= 2e-253) {
		tmp = (l_m * l_m) / (Math.pow((k * t_m), 2.0) * t_m);
	} else if (t_2 <= 2e+275) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * Math.sin(k)) * Math.tan(k)) * 2.0);
	} else {
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	t_2 = 2.0 / ((((math.pow(t_m, 3.0) / (l_m * l_m)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	tmp = 0
	if t_2 <= 2e-253:
		tmp = (l_m * l_m) / (math.pow((k * t_m), 2.0) * t_m)
	elif t_2 <= 2e+275:
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * math.sin(k)) * math.tan(k)) * 2.0)
	else:
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)))
	tmp = 0.0
	if (t_2 <= 2e-253)
		tmp = Float64(Float64(l_m * l_m) / Float64((Float64(k * t_m) ^ 2.0) * t_m));
	elseif (t_2 <= 2e+275)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * t_m) * Float64(k * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	t_2 = 2.0 / (((((t_m ^ 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	tmp = 0.0;
	if (t_2 <= 2e-253)
		tmp = (l_m * l_m) / (((k * t_m) ^ 2.0) * t_m);
	elseif (t_2 <= 2e+275)
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * sin(k)) * tan(k)) * 2.0);
	else
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := Block[{t$95$2 = N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 2e-253], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+275], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 2.0000000000000001e-253

   1. Initial program 79.0%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 68.3

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

   5. Applied rewrites 68.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 68.3

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

   7. Applied rewrites 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. pow-prod-down N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-*.f64 77.7

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

   9. Applied rewrites 77.7%

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

   if 2.0000000000000001e-253 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 1.99999999999999992e275

   1. Initial program 95.6%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 42.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 42.7

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

   7. Applied rewrites 42.7%

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

   if 1.99999999999999992e275 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 18.5%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 40.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 36.3%

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

   9. Taylor expanded in t around 0

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

       1. pow2 N/A

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

       2. pow2 N/A

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

       3. times-frac N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-/.f64 55.6

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

   11. Applied rewrites 55.6%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{-253}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t}\\ \mathbf{elif}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{t\_m}^{3}}{l\_m \cdot l\_m}\\ t_3 := \frac{2}{\left(\left(t\_2 \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-253}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{{\left(k \cdot t\_m\right)}^{2} \cdot t\_m}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{2}{\left(\left(t\_2 \cdot k\right) \cdot \tan k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot t\_m\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (/ (pow t_m 3.0) (* l_m l_m)))
        (t_3
         (/
          2.0
          (*
           (* (* t_2 (sin k)) (tan k))
           (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))))
   (*
    t_s
    (if (<= t_3 2e-253)
      (/ (* l_m l_m) (* (pow (* k t_m) 2.0) t_m))
      (if (<= t_3 2e+275)
        (/ 2.0 (* (* (* t_2 k) (tan k)) 2.0))
        (/ 2.0 (* (* (* (/ k l_m) (/ k l_m)) t_m) (* k k))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = pow(t_m, 3.0) / (l_m * l_m);
	double t_3 = 2.0 / (((t_2 * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	double tmp;
	if (t_3 <= 2e-253) {
		tmp = (l_m * l_m) / (pow((k * t_m), 2.0) * t_m);
	} else if (t_3 <= 2e+275) {
		tmp = 2.0 / (((t_2 * k) * tan(k)) * 2.0);
	} else {
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = (t_m ** 3.0d0) / (l_m * l_m)
    t_3 = 2.0d0 / (((t_2 * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    if (t_3 <= 2d-253) then
        tmp = (l_m * l_m) / (((k * t_m) ** 2.0d0) * t_m)
    else if (t_3 <= 2d+275) then
        tmp = 2.0d0 / (((t_2 * k) * tan(k)) * 2.0d0)
    else
        tmp = 2.0d0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = Math.pow(t_m, 3.0) / (l_m * l_m);
	double t_3 = 2.0 / (((t_2 * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	double tmp;
	if (t_3 <= 2e-253) {
		tmp = (l_m * l_m) / (Math.pow((k * t_m), 2.0) * t_m);
	} else if (t_3 <= 2e+275) {
		tmp = 2.0 / (((t_2 * k) * Math.tan(k)) * 2.0);
	} else {
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	t_2 = math.pow(t_m, 3.0) / (l_m * l_m)
	t_3 = 2.0 / (((t_2 * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	tmp = 0
	if t_3 <= 2e-253:
		tmp = (l_m * l_m) / (math.pow((k * t_m), 2.0) * t_m)
	elif t_3 <= 2e+275:
		tmp = 2.0 / (((t_2 * k) * math.tan(k)) * 2.0)
	else:
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64((t_m ^ 3.0) / Float64(l_m * l_m))
	t_3 = Float64(2.0 / Float64(Float64(Float64(t_2 * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)))
	tmp = 0.0
	if (t_3 <= 2e-253)
		tmp = Float64(Float64(l_m * l_m) / Float64((Float64(k * t_m) ^ 2.0) * t_m));
	elseif (t_3 <= 2e+275)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * k) * tan(k)) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * t_m) * Float64(k * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	t_2 = (t_m ^ 3.0) / (l_m * l_m);
	t_3 = 2.0 / (((t_2 * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	tmp = 0.0;
	if (t_3 <= 2e-253)
		tmp = (l_m * l_m) / (((k * t_m) ^ 2.0) * t_m);
	elseif (t_3 <= 2e+275)
		tmp = 2.0 / (((t_2 * k) * tan(k)) * 2.0);
	else
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := Block[{t$95$2 = N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 / N[(N[(N[(t$95$2 * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, 2e-253], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+275], N[(2.0 / N[(N[(N[(t$95$2 * k), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 2.0000000000000001e-253

   1. Initial program 79.0%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 68.3

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

   5. Applied rewrites 68.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 68.3

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

   7. Applied rewrites 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. pow-prod-down N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-*.f64 77.7

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

   9. Applied rewrites 77.7%

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

   if 2.0000000000000001e-253 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 1.99999999999999992e275

   1. Initial program 95.6%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 42.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 41.5%

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

   if 1.99999999999999992e275 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 18.5%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 40.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 36.3%

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

   9. Taylor expanded in t around 0

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

       1. pow2 N/A

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

       2. pow2 N/A

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

       3. times-frac N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-/.f64 55.6

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

   11. Applied rewrites 55.6%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{-253}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t}\\ \mathbf{elif}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot k\right) \cdot \tan k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\frac{{\log t\_m}^{2} \cdot 9 - {\log l\_m}^{2} \cdot 4}{\log t\_m \cdot 3 + \log l\_m \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9e-6)
    (/ 2.0 (* (* (/ k l_m) (/ k l_m)) (/ (* t_m (pow (sin k) 2.0)) (cos k))))
    (/
     2.0
     (*
      (*
       (*
        (exp
         (/
          (- (* (pow (log t_m) 2.0) 9.0) (* (pow (log l_m) 2.0) 4.0))
          (+ (* (log t_m) 3.0) (* (log l_m) 2.0))))
        (sin k))
       (tan k))
      (+ (+ 1.0 (* (/ k t_m) (/ k t_m))) 1.0))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 9e-6) {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * pow(sin(k), 2.0)) / cos(k)));
	} else {
		tmp = 2.0 / (((exp((((pow(log(t_m), 2.0) * 9.0) - (pow(log(l_m), 2.0) * 4.0)) / ((log(t_m) * 3.0) + (log(l_m) * 2.0)))) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 9d-6) then
        tmp = 2.0d0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ** 2.0d0)) / cos(k)))
    else
        tmp = 2.0d0 / (((exp(((((log(t_m) ** 2.0d0) * 9.0d0) - ((log(l_m) ** 2.0d0) * 4.0d0)) / ((log(t_m) * 3.0d0) + (log(l_m) * 2.0d0)))) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) * (k / t_m))) + 1.0d0))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 9e-6) {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(k)));
	} else {
		tmp = 2.0 / (((Math.exp((((Math.pow(Math.log(t_m), 2.0) * 9.0) - (Math.pow(Math.log(l_m), 2.0) * 4.0)) / ((Math.log(t_m) * 3.0) + (Math.log(l_m) * 2.0)))) * Math.sin(k)) * Math.tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 9e-6:
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * math.pow(math.sin(k), 2.0)) / math.cos(k)))
	else:
		tmp = 2.0 / (((math.exp((((math.pow(math.log(t_m), 2.0) * 9.0) - (math.pow(math.log(l_m), 2.0) * 4.0)) / ((math.log(t_m) * 3.0) + (math.log(l_m) * 2.0)))) * math.sin(k)) * math.tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 9e-6)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(Float64((log(t_m) ^ 2.0) * 9.0) - Float64((log(l_m) ^ 2.0) * 4.0)) / Float64(Float64(log(t_m) * 3.0) + Float64(log(l_m) * 2.0)))) * sin(k)) * tan(k)) * Float64(Float64(1.0 + Float64(Float64(k / t_m) * Float64(k / t_m))) + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 9e-6)
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ^ 2.0)) / cos(k)));
	else
		tmp = 2.0 / (((exp(((((log(t_m) ^ 2.0) * 9.0) - ((log(l_m) ^ 2.0) * 4.0)) / ((log(t_m) * 3.0) + (log(l_m) * 2.0)))) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9e-6], N[(2.0 / N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[(N[Power[N[Log[t$95$m], $MachinePrecision], 2.0], $MachinePrecision] * 9.0), $MachinePrecision] - N[(N[Power[N[Log[l$95$m], $MachinePrecision], 2.0], $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.00000000000000023e-6

   1. Initial program 48.7%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 55.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 55.0

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

   8. Applied rewrites 55.0%

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

   9. Taylor expanded in t around 0

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

       1. times-frac N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. pow2 N/A

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

       5. times-frac N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-pow.f64 N/A

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

       12. lift-sin.f64 N/A

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

       13. lower-cos.f64 72.8

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

   11. Applied rewrites 72.8%

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

   if 9.00000000000000023e-6 < t

   1. Initial program 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 42.3

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

   4. Applied rewrites 42.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 42.3

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

   6. Applied rewrites 42.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

      8. flip-- N/A

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

      9. lower-/.f64 N/A

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

   8. Applied rewrites 42.4%

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

   10. Applied rewrites 42.3%

       \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\frac{{\log t}^{2} \cdot 9 - {\log \ell}^{2} \cdot 4}{\log t \cdot 3 + \log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\frac{{\log t}^{2} \cdot 9 - {\log \ell}^{2} \cdot 4}{\log t \cdot 3 + \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot t\_2} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{{t\_m}^{3}}{l\_m}}{l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
   (*
    t_s
    (if (<=
         (/ 2.0 (* (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k)) t_2))
         2e+275)
      (/ 2.0 (* (* (* (/ (/ (pow t_m 3.0) l_m) l_m) (sin k)) (tan k)) t_2))
      (/
       2.0
       (* (* (/ k l_m) (/ k l_m)) (/ (* t_m (pow (sin k) 2.0)) (cos k))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = (1.0 + pow((k / t_m), 2.0)) + 1.0;
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * t_2)) <= 2e+275) {
		tmp = 2.0 / (((((pow(t_m, 3.0) / l_m) / l_m) * sin(k)) * tan(k)) * t_2);
	} else {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * pow(sin(k), 2.0)) / cos(k)));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l_m * l_m)) * sin(k)) * tan(k)) * t_2)) <= 2d+275) then
        tmp = 2.0d0 / ((((((t_m ** 3.0d0) / l_m) / l_m) * sin(k)) * tan(k)) * t_2)
    else
        tmp = 2.0d0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ** 2.0d0)) / cos(k)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = (1.0 + Math.pow((k / t_m), 2.0)) + 1.0;
	double tmp;
	if ((2.0 / ((((Math.pow(t_m, 3.0) / (l_m * l_m)) * Math.sin(k)) * Math.tan(k)) * t_2)) <= 2e+275) {
		tmp = 2.0 / (((((Math.pow(t_m, 3.0) / l_m) / l_m) * Math.sin(k)) * Math.tan(k)) * t_2);
	} else {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(k)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	t_2 = (1.0 + math.pow((k / t_m), 2.0)) + 1.0
	tmp = 0
	if (2.0 / ((((math.pow(t_m, 3.0) / (l_m * l_m)) * math.sin(k)) * math.tan(k)) * t_2)) <= 2e+275:
		tmp = 2.0 / (((((math.pow(t_m, 3.0) / l_m) / l_m) * math.sin(k)) * math.tan(k)) * t_2)
	else:
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * math.pow(math.sin(k), 2.0)) / math.cos(k)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * t_2)) <= 2e+275)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64((t_m ^ 3.0) / l_m) / l_m) * sin(k)) * tan(k)) * t_2));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	t_2 = (1.0 + ((k / t_m) ^ 2.0)) + 1.0;
	tmp = 0.0;
	if ((2.0 / (((((t_m ^ 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * t_2)) <= 2e+275)
		tmp = 2.0 / ((((((t_m ^ 3.0) / l_m) / l_m) * sin(k)) * tan(k)) * t_2);
	else
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ^ 2.0)) / cos(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := Block[{t$95$2 = N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2e+275], N[(2.0 / N[(N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 1.99999999999999992e275

   1. Initial program 80.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-pow.f64 83.4

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

   4. Applied rewrites 83.4%

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

   if 1.99999999999999992e275 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 18.5%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 40.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.0

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

   8. Applied rewrites 45.0%

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

   9. Taylor expanded in t around 0

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

       1. times-frac N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. pow2 N/A

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

       5. times-frac N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-pow.f64 N/A

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

       12. lift-sin.f64 N/A

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

       13. lower-cos.f64 77.0

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

   11. Applied rewrites 77.0%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + t\_2\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{t\_m}^{3}}{l\_m}}{l\_m} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(t\_2 + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<=
         (/
          2.0
          (*
           (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k))
           (+ (+ 1.0 t_2) 1.0)))
         2e+275)
      (/
       2.0
       (* (* (/ (/ (pow t_m 3.0) l_m) l_m) (sin k)) (* (tan k) (+ t_2 2.0))))
      (/
       2.0
       (* (* (/ k l_m) (/ k l_m)) (/ (* t_m (pow (sin k) 2.0)) (cos k))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + t_2) + 1.0))) <= 2e+275) {
		tmp = 2.0 / ((((pow(t_m, 3.0) / l_m) / l_m) * sin(k)) * (tan(k) * (t_2 + 2.0)));
	} else {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * pow(sin(k), 2.0)) / cos(k)));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (k / t_m) ** 2.0d0
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0d0 + t_2) + 1.0d0))) <= 2d+275) then
        tmp = 2.0d0 / (((((t_m ** 3.0d0) / l_m) / l_m) * sin(k)) * (tan(k) * (t_2 + 2.0d0)))
    else
        tmp = 2.0d0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ** 2.0d0)) / cos(k)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if ((2.0 / ((((Math.pow(t_m, 3.0) / (l_m * l_m)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + t_2) + 1.0))) <= 2e+275) {
		tmp = 2.0 / ((((Math.pow(t_m, 3.0) / l_m) / l_m) * Math.sin(k)) * (Math.tan(k) * (t_2 + 2.0)));
	} else {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(k)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	t_2 = math.pow((k / t_m), 2.0)
	tmp = 0
	if (2.0 / ((((math.pow(t_m, 3.0) / (l_m * l_m)) * math.sin(k)) * math.tan(k)) * ((1.0 + t_2) + 1.0))) <= 2e+275:
		tmp = 2.0 / ((((math.pow(t_m, 3.0) / l_m) / l_m) * math.sin(k)) * (math.tan(k) * (t_2 + 2.0)))
	else:
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * math.pow(math.sin(k), 2.0)) / math.cos(k)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + t_2) + 1.0))) <= 2e+275)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / l_m) / l_m) * sin(k)) * Float64(tan(k) * Float64(t_2 + 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	t_2 = (k / t_m) ^ 2.0;
	tmp = 0.0;
	if ((2.0 / (((((t_m ^ 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + t_2) + 1.0))) <= 2e+275)
		tmp = 2.0 / (((((t_m ^ 3.0) / l_m) / l_m) * sin(k)) * (tan(k) * (t_2 + 2.0)));
	else
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ^ 2.0)) / cos(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+275], N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$2 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 1.99999999999999992e275

   1. Initial program 80.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 15.6

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

   4. Applied rewrites 15.6%

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

   6. Applied rewrites 83.4%

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

   if 1.99999999999999992e275 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 18.5%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 40.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.0

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

   8. Applied rewrites 45.0%

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

   9. Taylor expanded in t around 0

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

       1. times-frac N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. pow2 N/A

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

       5. times-frac N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-pow.f64 N/A

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

       12. lift-sin.f64 N/A

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

       13. lower-cos.f64 77.0

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

   11. Applied rewrites 77.0%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot t\_2} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{2}{\left(\left({t\_m}^{3} \cdot \frac{\sin k}{l\_m \cdot l\_m}\right) \cdot \tan k\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
   (*
    t_s
    (if (<=
         (/ 2.0 (* (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k)) t_2))
         2e+275)
      (/ 2.0 (* (* (* (pow t_m 3.0) (/ (sin k) (* l_m l_m))) (tan k)) t_2))
      (/
       2.0
       (* (* (/ k l_m) (/ k l_m)) (/ (* t_m (pow (sin k) 2.0)) (cos k))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = (1.0 + pow((k / t_m), 2.0)) + 1.0;
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * t_2)) <= 2e+275) {
		tmp = 2.0 / (((pow(t_m, 3.0) * (sin(k) / (l_m * l_m))) * tan(k)) * t_2);
	} else {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * pow(sin(k), 2.0)) / cos(k)));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l_m * l_m)) * sin(k)) * tan(k)) * t_2)) <= 2d+275) then
        tmp = 2.0d0 / ((((t_m ** 3.0d0) * (sin(k) / (l_m * l_m))) * tan(k)) * t_2)
    else
        tmp = 2.0d0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ** 2.0d0)) / cos(k)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = (1.0 + Math.pow((k / t_m), 2.0)) + 1.0;
	double tmp;
	if ((2.0 / ((((Math.pow(t_m, 3.0) / (l_m * l_m)) * Math.sin(k)) * Math.tan(k)) * t_2)) <= 2e+275) {
		tmp = 2.0 / (((Math.pow(t_m, 3.0) * (Math.sin(k) / (l_m * l_m))) * Math.tan(k)) * t_2);
	} else {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(k)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	t_2 = (1.0 + math.pow((k / t_m), 2.0)) + 1.0
	tmp = 0
	if (2.0 / ((((math.pow(t_m, 3.0) / (l_m * l_m)) * math.sin(k)) * math.tan(k)) * t_2)) <= 2e+275:
		tmp = 2.0 / (((math.pow(t_m, 3.0) * (math.sin(k) / (l_m * l_m))) * math.tan(k)) * t_2)
	else:
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * math.pow(math.sin(k), 2.0)) / math.cos(k)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * t_2)) <= 2e+275)
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 3.0) * Float64(sin(k) / Float64(l_m * l_m))) * tan(k)) * t_2));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	t_2 = (1.0 + ((k / t_m) ^ 2.0)) + 1.0;
	tmp = 0.0;
	if ((2.0 / (((((t_m ^ 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * t_2)) <= 2e+275)
		tmp = 2.0 / ((((t_m ^ 3.0) * (sin(k) / (l_m * l_m))) * tan(k)) * t_2);
	else
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ^ 2.0)) / cos(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := Block[{t$95$2 = N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2e+275], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 1.99999999999999992e275

   1. Initial program 80.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 82.0

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

   4. Applied rewrites 82.0%

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

   if 1.99999999999999992e275 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 18.5%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 40.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.0

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

   8. Applied rewrites 45.0%

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

   9. Taylor expanded in t around 0

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

       1. times-frac N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. pow2 N/A

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

       5. times-frac N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-pow.f64 N/A

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

       12. lift-sin.f64 N/A

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

       13. lower-cos.f64 77.0

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

   11. Applied rewrites 77.0%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{2}{\left(\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \frac{1}{{\sin k}^{-1}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9e-6)
    (/ 2.0 (* (* (/ k l_m) (/ k l_m)) (/ (* t_m (pow (sin k) 2.0)) (cos k))))
    (/
     2.0
     (*
      (*
       (*
        (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0)))
        (/ 1.0 (pow (sin k) -1.0)))
       (tan k))
      (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 9e-6) {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * pow(sin(k), 2.0)) / cos(k)));
	} else {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * (1.0 / pow(sin(k), -1.0))) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 9d-6) then
        tmp = 2.0d0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ** 2.0d0)) / cos(k)))
    else
        tmp = 2.0d0 / (((exp(((log(t_m) * 3.0d0) - (log(l_m) * 2.0d0))) * (1.0d0 / (sin(k) ** (-1.0d0)))) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 9e-6) {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(k)));
	} else {
		tmp = 2.0 / (((Math.exp(((Math.log(t_m) * 3.0) - (Math.log(l_m) * 2.0))) * (1.0 / Math.pow(Math.sin(k), -1.0))) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 9e-6:
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * math.pow(math.sin(k), 2.0)) / math.cos(k)))
	else:
		tmp = 2.0 / (((math.exp(((math.log(t_m) * 3.0) - (math.log(l_m) * 2.0))) * (1.0 / math.pow(math.sin(k), -1.0))) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 9e-6)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * Float64(1.0 / (sin(k) ^ -1.0))) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 9e-6)
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ^ 2.0)) / cos(k)));
	else
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * (1.0 / (sin(k) ^ -1.0))) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9e-6], N[(2.0 / N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Power[N[Sin[k], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.00000000000000023e-6

   1. Initial program 48.7%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 55.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 55.0

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

   8. Applied rewrites 55.0%

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

   9. Taylor expanded in t around 0

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

       1. times-frac N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. pow2 N/A

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

       5. times-frac N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-pow.f64 N/A

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

       12. lift-sin.f64 N/A

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

       13. lower-cos.f64 72.8

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

   11. Applied rewrites 72.8%

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

   if 9.00000000000000023e-6 < t

   1. Initial program 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 42.3

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

   4. Applied rewrites 42.3%

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

      1. lift-sin.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. pow-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-sin.f64 42.3

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

   6. Applied rewrites 42.3%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \frac{1}{{\sin k}^{-1}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\_m\right)}^{2} \cdot \left(\left(k \cdot k\right) \cdot 0.3333333333333333 + 2\right) + {k}^{4}}{l\_m \cdot l\_m} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot t\_m\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       2e+275)
    (/
     2.0
     (*
      (/
       (+
        (* (pow (* k t_m) 2.0) (+ (* (* k k) 0.3333333333333333) 2.0))
        (pow k 4.0))
       (* l_m l_m))
      t_m))
    (/ 2.0 (* (* (* (/ k l_m) (/ k l_m)) t_m) (* k k))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= 2e+275) {
		tmp = 2.0 / ((((pow((k * t_m), 2.0) * (((k * k) * 0.3333333333333333) + 2.0)) + pow(k, 4.0)) / (l_m * l_m)) * t_m);
	} else {
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))) <= 2d+275) then
        tmp = 2.0d0 / ((((((k * t_m) ** 2.0d0) * (((k * k) * 0.3333333333333333d0) + 2.0d0)) + (k ** 4.0d0)) / (l_m * l_m)) * t_m)
    else
        tmp = 2.0d0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if ((2.0 / ((((Math.pow(t_m, 3.0) / (l_m * l_m)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0))) <= 2e+275) {
		tmp = 2.0 / ((((Math.pow((k * t_m), 2.0) * (((k * k) * 0.3333333333333333) + 2.0)) + Math.pow(k, 4.0)) / (l_m * l_m)) * t_m);
	} else {
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if (2.0 / ((((math.pow(t_m, 3.0) / (l_m * l_m)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))) <= 2e+275:
		tmp = 2.0 / ((((math.pow((k * t_m), 2.0) * (((k * k) * 0.3333333333333333) + 2.0)) + math.pow(k, 4.0)) / (l_m * l_m)) * t_m)
	else:
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))) <= 2e+275)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * Float64(Float64(Float64(k * k) * 0.3333333333333333) + 2.0)) + (k ^ 4.0)) / Float64(l_m * l_m)) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * t_m) * Float64(k * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if ((2.0 / (((((t_m ^ 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0))) <= 2e+275)
		tmp = 2.0 / ((((((k * t_m) ^ 2.0) * (((k * k) * 0.3333333333333333) + 2.0)) + (k ^ 4.0)) / (l_m * l_m)) * t_m);
	else
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+275], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 1.99999999999999992e275

   1. Initial program 80.1%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 65.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(2 + \frac{1}{3} \cdot {k}^{2}\right)\right)}{{\ell}^{2}} + \frac{{k}^{4}}{{\ell}^{2}}\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(2 + \frac{1}{3} \cdot {k}^{2}\right)\right)}{{\ell}^{2}} + \frac{{k}^{4}}{{\ell}^{2}}\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(2 + \frac{1}{3} \cdot {k}^{2}\right)\right)}{{\ell}^{2}} + \frac{{k}^{4}}{{\ell}^{2}}\right) \cdot t} \]

   8. Applied rewrites 76.7%

      \[\leadsto \frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot \left(\left(k \cdot k\right) \cdot 0.3333333333333333 + 2\right) + {k}^{4}}{\ell \cdot \ell} \cdot \color{blue}{t}} \]

   if 1.99999999999999992e275 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 18.5%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 40.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 36.3%

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

   9. Taylor expanded in t around 0

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

       1. pow2 N/A

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

       2. pow2 N/A

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

       3. times-frac N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-/.f64 55.6

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

   11. Applied rewrites 55.6%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot \left(\left(k \cdot k\right) \cdot 0.3333333333333333 + 2\right) + {k}^{4}}{\ell \cdot \ell} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 84.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\right) \cdot \frac{\sin k}{\cos k}\right) \cdot \left(\left(1 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9e-6)
    (/ 2.0 (* (* (/ k l_m) (/ k l_m)) (/ (* t_m (pow (sin k) 2.0)) (cos k))))
    (/
     2.0
     (*
      (*
       (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (sin k))
       (/ (sin k) (cos k)))
      (+ (+ 1.0 (* (/ k t_m) (/ k t_m))) 1.0))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 9e-6) {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * pow(sin(k), 2.0)) / cos(k)));
	} else {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * (sin(k) / cos(k))) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 9d-6) then
        tmp = 2.0d0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ** 2.0d0)) / cos(k)))
    else
        tmp = 2.0d0 / (((exp(((log(t_m) * 3.0d0) - (log(l_m) * 2.0d0))) * sin(k)) * (sin(k) / cos(k))) * ((1.0d0 + ((k / t_m) * (k / t_m))) + 1.0d0))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 9e-6) {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(k)));
	} else {
		tmp = 2.0 / (((Math.exp(((Math.log(t_m) * 3.0) - (Math.log(l_m) * 2.0))) * Math.sin(k)) * (Math.sin(k) / Math.cos(k))) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 9e-6:
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * math.pow(math.sin(k), 2.0)) / math.cos(k)))
	else:
		tmp = 2.0 / (((math.exp(((math.log(t_m) * 3.0) - (math.log(l_m) * 2.0))) * math.sin(k)) * (math.sin(k) / math.cos(k))) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 9e-6)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * sin(k)) * Float64(sin(k) / cos(k))) * Float64(Float64(1.0 + Float64(Float64(k / t_m) * Float64(k / t_m))) + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 9e-6)
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ^ 2.0)) / cos(k)));
	else
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * (sin(k) / cos(k))) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9e-6], N[(2.0 / N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.00000000000000023e-6

   1. Initial program 48.7%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 55.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 55.0

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

   8. Applied rewrites 55.0%

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

   9. Taylor expanded in t around 0

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

       1. times-frac N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. pow2 N/A

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

       5. times-frac N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-pow.f64 N/A

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

       12. lift-sin.f64 N/A

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

       13. lower-cos.f64 72.8

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

   11. Applied rewrites 72.8%

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

   if 9.00000000000000023e-6 < t

   1. Initial program 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 42.3

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

   4. Applied rewrites 42.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 42.3

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

   6. Applied rewrites 42.3%

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

      1. lift-tan.f64 N/A

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

      2. tan-quot N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-cos.f64 42.3

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

   8. Applied rewrites 42.3%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \frac{\sin k}{\cos k}\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 68.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{{\left(k \cdot t\_m\right)}^{2} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot t\_m\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       2e+275)
    (/ (* l_m l_m) (* (pow (* k t_m) 2.0) t_m))
    (/ 2.0 (* (* (* (/ k l_m) (/ k l_m)) t_m) (* k k))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= 2e+275) {
		tmp = (l_m * l_m) / (pow((k * t_m), 2.0) * t_m);
	} else {
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))) <= 2d+275) then
        tmp = (l_m * l_m) / (((k * t_m) ** 2.0d0) * t_m)
    else
        tmp = 2.0d0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if ((2.0 / ((((Math.pow(t_m, 3.0) / (l_m * l_m)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0))) <= 2e+275) {
		tmp = (l_m * l_m) / (Math.pow((k * t_m), 2.0) * t_m);
	} else {
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if (2.0 / ((((math.pow(t_m, 3.0) / (l_m * l_m)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))) <= 2e+275:
		tmp = (l_m * l_m) / (math.pow((k * t_m), 2.0) * t_m)
	else:
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))) <= 2e+275)
		tmp = Float64(Float64(l_m * l_m) / Float64((Float64(k * t_m) ^ 2.0) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * t_m) * Float64(k * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if ((2.0 / (((((t_m ^ 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0))) <= 2e+275)
		tmp = (l_m * l_m) / (((k * t_m) ^ 2.0) * t_m);
	else
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+275], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 1.99999999999999992e275

   1. Initial program 80.1%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 64.6

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

   5. Applied rewrites 64.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 64.5

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

   7. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. pow-prod-down N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-*.f64 73.4

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

   9. Applied rewrites 73.4%

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

   if 1.99999999999999992e275 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 18.5%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 40.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 36.3%

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

   9. Taylor expanded in t around 0

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

       1. pow2 N/A

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

       2. pow2 N/A

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

       3. times-frac N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-/.f64 55.6

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

   11. Applied rewrites 55.6%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 66.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{k \cdot \left({t\_m}^{3} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot t\_m\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       2e+275)
    (/ (* l_m l_m) (* k (* (pow t_m 3.0) k)))
    (/ 2.0 (* (* (* (/ k l_m) (/ k l_m)) t_m) (* k k))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= 2e+275) {
		tmp = (l_m * l_m) / (k * (pow(t_m, 3.0) * k));
	} else {
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))) <= 2d+275) then
        tmp = (l_m * l_m) / (k * ((t_m ** 3.0d0) * k))
    else
        tmp = 2.0d0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if ((2.0 / ((((Math.pow(t_m, 3.0) / (l_m * l_m)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0))) <= 2e+275) {
		tmp = (l_m * l_m) / (k * (Math.pow(t_m, 3.0) * k));
	} else {
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if (2.0 / ((((math.pow(t_m, 3.0) / (l_m * l_m)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))) <= 2e+275:
		tmp = (l_m * l_m) / (k * (math.pow(t_m, 3.0) * k))
	else:
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))) <= 2e+275)
		tmp = Float64(Float64(l_m * l_m) / Float64(k * Float64((t_m ^ 3.0) * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * t_m) * Float64(k * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if ((2.0 / (((((t_m ^ 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0))) <= 2e+275)
		tmp = (l_m * l_m) / (k * ((t_m ^ 3.0) * k));
	else
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+275], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(k * N[(N[Power[t$95$m, 3.0], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 1.99999999999999992e275

   1. Initial program 80.1%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 64.6

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

   5. Applied rewrites 64.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 71.5

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

   7. Applied rewrites 71.5%

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

   if 1.99999999999999992e275 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 18.5%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 40.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 36.3%

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

   9. Taylor expanded in t around 0

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

       1. pow2 N/A

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

       2. pow2 N/A

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

       3. times-frac N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-/.f64 55.6

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

   11. Applied rewrites 55.6%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left({t}^{3} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 84.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9e-6)
    (/ 2.0 (* (* (/ k l_m) (/ k l_m)) (/ (* t_m (pow (sin k) 2.0)) (cos k))))
    (/
     2.0
     (*
      (* (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (sin k)) (tan k))
      (+ (+ 1.0 (* (/ k t_m) (/ k t_m))) 1.0))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 9e-6) {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * pow(sin(k), 2.0)) / cos(k)));
	} else {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 9d-6) then
        tmp = 2.0d0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ** 2.0d0)) / cos(k)))
    else
        tmp = 2.0d0 / (((exp(((log(t_m) * 3.0d0) - (log(l_m) * 2.0d0))) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) * (k / t_m))) + 1.0d0))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 9e-6) {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(k)));
	} else {
		tmp = 2.0 / (((Math.exp(((Math.log(t_m) * 3.0) - (Math.log(l_m) * 2.0))) * Math.sin(k)) * Math.tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 9e-6:
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * math.pow(math.sin(k), 2.0)) / math.cos(k)))
	else:
		tmp = 2.0 / (((math.exp(((math.log(t_m) * 3.0) - (math.log(l_m) * 2.0))) * math.sin(k)) * math.tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 9e-6)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * sin(k)) * tan(k)) * Float64(Float64(1.0 + Float64(Float64(k / t_m) * Float64(k / t_m))) + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 9e-6)
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ^ 2.0)) / cos(k)));
	else
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9e-6], N[(2.0 / N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.00000000000000023e-6

   1. Initial program 48.7%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 55.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)} \]

      7. lift-*.f64 55.0

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)} \]

   8. Applied rewrites 55.0%

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   10. Step-by-step derivation

       1. times-frac N/A

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

       3. pow2 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

       4. pow2 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

       5. times-frac N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\cos k}} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\cos k}} \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

       9. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\cos k}}} \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos \color{blue}{k}}} \]

       11. lower-pow.f64 N/A

           \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

       12. lift-sin.f64 N/A

           \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

       13. lower-cos.f64 72.8

           \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

   11. Applied rewrites 72.8%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

   if 9.00000000000000023e-6 < t

   1. Initial program 67.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t} \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-log.f64 42.3

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 42.3%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\color{blue}{\left(\frac{k}{t}\right)}}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right) + 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t}} \cdot \frac{k}{t}\right) + 1\right)} \]

      6. lift-/.f64 42.3

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \color{blue}{\frac{k}{t}}\right) + 1\right)} \]

   6. Applied rewrites 42.3%

      \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 83.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + k \cdot \frac{k}{t\_m \cdot t\_m}\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9e-6)
    (/ 2.0 (* (* (/ k l_m) (/ k l_m)) (/ (* t_m (pow (sin k) 2.0)) (cos k))))
    (/
     2.0
     (*
      (* (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (sin k)) (tan k))
      (+ (+ 1.0 (* k (/ k (* t_m t_m)))) 1.0))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 9e-6) {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * pow(sin(k), 2.0)) / cos(k)));
	} else {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * tan(k)) * ((1.0 + (k * (k / (t_m * t_m)))) + 1.0));
	}
	return t_s * tmp;
}

Fortran

l_m =     private
t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 9d-6) then
        tmp = 2.0d0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ** 2.0d0)) / cos(k)))
    else
        tmp = 2.0d0 / (((exp(((log(t_m) * 3.0d0) - (log(l_m) * 2.0d0))) * sin(k)) * tan(k)) * ((1.0d0 + (k * (k / (t_m * t_m)))) + 1.0d0))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 9e-6) {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(k)));
	} else {
		tmp = 2.0 / (((Math.exp(((Math.log(t_m) * 3.0) - (Math.log(l_m) * 2.0))) * Math.sin(k)) * Math.tan(k)) * ((1.0 + (k * (k / (t_m * t_m)))) + 1.0));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 9e-6:
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * math.pow(math.sin(k), 2.0)) / math.cos(k)))
	else:
		tmp = 2.0 / (((math.exp(((math.log(t_m) * 3.0) - (math.log(l_m) * 2.0))) * math.sin(k)) * math.tan(k)) * ((1.0 + (k * (k / (t_m * t_m)))) + 1.0))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 9e-6)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * sin(k)) * tan(k)) * Float64(Float64(1.0 + Float64(k * Float64(k / Float64(t_m * t_m)))) + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 9e-6)
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ^ 2.0)) / cos(k)));
	else
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * tan(k)) * ((1.0 + (k * (k / (t_m * t_m)))) + 1.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9e-6], N[(2.0 / N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.00000000000000023e-6

   1. Initial program 48.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

   5. Applied rewrites 55.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\color{blue}{k} \cdot k\right)} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)} \]

      7. lift-*.f64 55.0

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)} \]

   8. Applied rewrites 55.0%

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   10. Step-by-step derivation

       1. times-frac N/A

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

       3. pow2 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

       4. pow2 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

       5. times-frac N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\cos k}} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\cos k}} \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

       9. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\cos k}}} \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos \color{blue}{k}}} \]

       11. lower-pow.f64 N/A

           \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

       12. lift-sin.f64 N/A

           \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

       13. lower-cos.f64 72.8

           \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

   11. Applied rewrites 72.8%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

   if 9.00000000000000023e-6 < t

   1. Initial program 67.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t} \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-log.f64 42.3

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 42.3%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\color{blue}{\left(\frac{k}{t}\right)}}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right) + 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1\right)} \]

      4. times-frac N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{{t}^{2}}}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{k \cdot \frac{k}{{t}^{2}}}\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{k \cdot \frac{k}{{t}^{2}}}\right) + 1\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + k \cdot \color{blue}{\frac{k}{{t}^{2}}}\right) + 1\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + k \cdot \frac{k}{\color{blue}{t \cdot t}}\right) + 1\right)} \]

      10. lift-*.f64 42.3

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + k \cdot \frac{k}{\color{blue}{t \cdot t}}\right) + 1\right)} \]

   6. Applied rewrites 42.3%

      \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{k \cdot \frac{k}{t \cdot t}}\right) + 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + k \cdot \frac{k}{t \cdot t}\right) + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 72.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3800:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 3800.0)
    (/
     2.0
     (*
      (* (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (sin k)) (tan k))
      2.0))
    (/
     2.0
     (* (* (/ k l_m) (/ k l_m)) (/ (* t_m (pow (sin k) 2.0)) (cos k)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 3800.0) {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * tan(k)) * 2.0);
	} else {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * pow(sin(k), 2.0)) / cos(k)));
	}
	return t_s * tmp;
}

Fortran

l_m =     private
t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3800.0d0) then
        tmp = 2.0d0 / (((exp(((log(t_m) * 3.0d0) - (log(l_m) * 2.0d0))) * sin(k)) * tan(k)) * 2.0d0)
    else
        tmp = 2.0d0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ** 2.0d0)) / cos(k)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 3800.0) {
		tmp = 2.0 / (((Math.exp(((Math.log(t_m) * 3.0) - (Math.log(l_m) * 2.0))) * Math.sin(k)) * Math.tan(k)) * 2.0);
	} else {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(k)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 3800.0:
		tmp = 2.0 / (((math.exp(((math.log(t_m) * 3.0) - (math.log(l_m) * 2.0))) * math.sin(k)) * math.tan(k)) * 2.0)
	else:
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * math.pow(math.sin(k), 2.0)) / math.cos(k)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 3800.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * sin(k)) * tan(k)) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 3800.0)
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * tan(k)) * 2.0);
	else
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ^ 2.0)) / cos(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 3800.0], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3800

   1. Initial program 56.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t} \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-log.f64 19.0

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 19.0%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\color{blue}{\left(\frac{k}{t}\right)}}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right) + 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1\right)} \]

      4. times-frac N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{{t}^{2}}}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{k \cdot \frac{k}{{t}^{2}}}\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{k \cdot \frac{k}{{t}^{2}}}\right) + 1\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + k \cdot \color{blue}{\frac{k}{{t}^{2}}}\right) + 1\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + k \cdot \frac{k}{\color{blue}{t \cdot t}}\right) + 1\right)} \]

      10. lift-*.f64 16.5

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + k \cdot \frac{k}{\color{blue}{t \cdot t}}\right) + 1\right)} \]

   6. Applied rewrites 16.5%

      \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{k \cdot \frac{k}{t \cdot t}}\right) + 1\right)} \]

   7. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{2}} \]
   8. Step-by-step derivation

   9. Applied rewrites 18.5%

      \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{2}} \]

   if 3800 < k

   1. Initial program 44.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

   5. Applied rewrites 56.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\color{blue}{k} \cdot k\right)} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)} \]

      7. lift-*.f64 56.5

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)} \]

   8. Applied rewrites 56.5%

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   10. Step-by-step derivation

       1. times-frac N/A

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

       3. pow2 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

       4. pow2 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

       5. times-frac N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\cos k}} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\cos k}} \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

       9. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\cos k}}} \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos \color{blue}{k}}} \]

       11. lower-pow.f64 N/A

           \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

       12. lift-sin.f64 N/A

           \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

       13. lower-cos.f64 81.5

           \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

   11. Applied rewrites 81.5%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3800:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 63.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(t\_m \cdot t\_m\right) \cdot \left(2 + 0.3333333333333333 \cdot \left(k \cdot k\right)\right) + k \cdot k}{l\_m \cdot l\_m} \cdot t\_m\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot t\_m\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       2e+275)
    (/
     2.0
     (*
      (*
       (/
        (+ (* (* t_m t_m) (+ 2.0 (* 0.3333333333333333 (* k k)))) (* k k))
        (* l_m l_m))
       t_m)
      (* k k)))
    (/ 2.0 (* (* (* (/ k l_m) (/ k l_m)) t_m) (* k k))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= 2e+275) {
		tmp = 2.0 / ((((((t_m * t_m) * (2.0 + (0.3333333333333333 * (k * k)))) + (k * k)) / (l_m * l_m)) * t_m) * (k * k));
	} else {
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k));
	}
	return t_s * tmp;
}

Fortran

l_m =     private
t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))) <= 2d+275) then
        tmp = 2.0d0 / ((((((t_m * t_m) * (2.0d0 + (0.3333333333333333d0 * (k * k)))) + (k * k)) / (l_m * l_m)) * t_m) * (k * k))
    else
        tmp = 2.0d0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if ((2.0 / ((((Math.pow(t_m, 3.0) / (l_m * l_m)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0))) <= 2e+275) {
		tmp = 2.0 / ((((((t_m * t_m) * (2.0 + (0.3333333333333333 * (k * k)))) + (k * k)) / (l_m * l_m)) * t_m) * (k * k));
	} else {
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if (2.0 / ((((math.pow(t_m, 3.0) / (l_m * l_m)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))) <= 2e+275:
		tmp = 2.0 / ((((((t_m * t_m) * (2.0 + (0.3333333333333333 * (k * k)))) + (k * k)) / (l_m * l_m)) * t_m) * (k * k))
	else:
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))) <= 2e+275)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * Float64(2.0 + Float64(0.3333333333333333 * Float64(k * k)))) + Float64(k * k)) / Float64(l_m * l_m)) * t_m) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * t_m) * Float64(k * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if ((2.0 / (((((t_m ^ 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0))) <= 2e+275)
		tmp = 2.0 / ((((((t_m * t_m) * (2.0 + (0.3333333333333333 * (k * k)))) + (k * k)) / (l_m * l_m)) * t_m) * (k * k));
	else
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+275], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(2.0 + N[(0.3333333333333333 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 1.99999999999999992e275

   1. Initial program 80.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

   5. Applied rewrites 65.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(t \cdot \left({t}^{2} \cdot \left(\frac{1}{3} \cdot \frac{{k}^{2}}{{\ell}^{2}} + 2 \cdot \frac{1}{{\ell}^{2}}\right) + \frac{{k}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left({t}^{2} \cdot \left(\frac{1}{3} \cdot \frac{{k}^{2}}{{\ell}^{2}} + 2 \cdot \frac{1}{{\ell}^{2}}\right) + \frac{{k}^{2}}{{\ell}^{2}}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({t}^{2} \cdot \left(\frac{1}{3} \cdot \frac{{k}^{2}}{{\ell}^{2}} + 2 \cdot \frac{1}{{\ell}^{2}}\right) + \frac{{k}^{2}}{{\ell}^{2}}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

   8. Applied rewrites 67.5%

      \[\leadsto \frac{2}{\left(\left(\left(\frac{\left(k \cdot k\right) \cdot 0.3333333333333333}{\ell \cdot \ell} - -2 \cdot {\ell}^{-2}\right) \cdot \left(t \cdot t\right) + \frac{k \cdot k}{\ell \cdot \ell}\right) \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

   9. Taylor expanded in l around 0

      \[\leadsto \frac{2}{\left(\frac{{t}^{2} \cdot \left(2 + \frac{1}{3} \cdot {k}^{2}\right) + {k}^{2}}{{\ell}^{2}} \cdot t\right) \cdot \left(k \cdot k\right)} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{{t}^{2} \cdot \left(2 + \frac{1}{3} \cdot {k}^{2}\right) + {k}^{2}}{{\ell}^{2}} \cdot t\right) \cdot \left(k \cdot k\right)} \]

       2. lower-+.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{{t}^{2} \cdot \left(2 + \frac{1}{3} \cdot {k}^{2}\right) + {k}^{2}}{{\ell}^{2}} \cdot t\right) \cdot \left(k \cdot k\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{{t}^{2} \cdot \left(2 + \frac{1}{3} \cdot {k}^{2}\right) + {k}^{2}}{{\ell}^{2}} \cdot t\right) \cdot \left(k \cdot k\right)} \]

       4. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \left(2 + \frac{1}{3} \cdot {k}^{2}\right) + {k}^{2}}{{\ell}^{2}} \cdot t\right) \cdot \left(k \cdot k\right)} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \left(2 + \frac{1}{3} \cdot {k}^{2}\right) + {k}^{2}}{{\ell}^{2}} \cdot t\right) \cdot \left(k \cdot k\right)} \]

       6. lower-+.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \left(2 + \frac{1}{3} \cdot {k}^{2}\right) + {k}^{2}}{{\ell}^{2}} \cdot t\right) \cdot \left(k \cdot k\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \left(2 + \frac{1}{3} \cdot {k}^{2}\right) + {k}^{2}}{{\ell}^{2}} \cdot t\right) \cdot \left(k \cdot k\right)} \]

       8. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \left(2 + \frac{1}{3} \cdot \left(k \cdot k\right)\right) + {k}^{2}}{{\ell}^{2}} \cdot t\right) \cdot \left(k \cdot k\right)} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \left(2 + \frac{1}{3} \cdot \left(k \cdot k\right)\right) + {k}^{2}}{{\ell}^{2}} \cdot t\right) \cdot \left(k \cdot k\right)} \]

       10. pow2 N/A

           \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \left(2 + \frac{1}{3} \cdot \left(k \cdot k\right)\right) + k \cdot k}{{\ell}^{2}} \cdot t\right) \cdot \left(k \cdot k\right)} \]

       11. lift-*.f64 N/A

           \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \left(2 + \frac{1}{3} \cdot \left(k \cdot k\right)\right) + k \cdot k}{{\ell}^{2}} \cdot t\right) \cdot \left(k \cdot k\right)} \]

       12. pow2 N/A

           \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \left(2 + \frac{1}{3} \cdot \left(k \cdot k\right)\right) + k \cdot k}{\ell \cdot \ell} \cdot t\right) \cdot \left(k \cdot k\right)} \]

       13. lift-*.f64 67.8

           \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \left(2 + 0.3333333333333333 \cdot \left(k \cdot k\right)\right) + k \cdot k}{\ell \cdot \ell} \cdot t\right) \cdot \left(k \cdot k\right)} \]

   11. Applied rewrites 67.8%

       \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \left(2 + 0.3333333333333333 \cdot \left(k \cdot k\right)\right) + k \cdot k}{\ell \cdot \ell} \cdot t\right) \cdot \left(k \cdot k\right)} \]

   if 1.99999999999999992e275 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 18.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

   5. Applied rewrites 40.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(t \cdot \left({t}^{2} \cdot \left(\frac{1}{3} \cdot \frac{{k}^{2}}{{\ell}^{2}} + 2 \cdot \frac{1}{{\ell}^{2}}\right) + \frac{{k}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left({t}^{2} \cdot \left(\frac{1}{3} \cdot \frac{{k}^{2}}{{\ell}^{2}} + 2 \cdot \frac{1}{{\ell}^{2}}\right) + \frac{{k}^{2}}{{\ell}^{2}}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({t}^{2} \cdot \left(\frac{1}{3} \cdot \frac{{k}^{2}}{{\ell}^{2}} + 2 \cdot \frac{1}{{\ell}^{2}}\right) + \frac{{k}^{2}}{{\ell}^{2}}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

   8. Applied rewrites 36.3%

      \[\leadsto \frac{2}{\left(\left(\left(\frac{\left(k \cdot k\right) \cdot 0.3333333333333333}{\ell \cdot \ell} - -2 \cdot {\ell}^{-2}\right) \cdot \left(t \cdot t\right) + \frac{k \cdot k}{\ell \cdot \ell}\right) \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot t\right) \cdot \left(k \cdot k\right)} \]
   10. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{{\ell}^{2}} \cdot t\right) \cdot \left(k \cdot k\right)} \]

       2. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot t\right) \cdot \left(k \cdot k\right)} \]

       3. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

       6. lower-/.f64 55.6

          \[\leadsto \frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

   11. Applied rewrites 55.6%

       \[\leadsto \frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \left(2 + 0.3333333333333333 \cdot \left(k \cdot k\right)\right) + k \cdot k}{\ell \cdot \ell} \cdot t\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 62.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{2}{\left(\left(2 \cdot \frac{t\_m \cdot t\_m}{l\_m \cdot l\_m}\right) \cdot t\_m\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot t\_m\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       2e+275)
    (/ 2.0 (* (* (* 2.0 (/ (* t_m t_m) (* l_m l_m))) t_m) (* k k)))
    (/ 2.0 (* (* (* (/ k l_m) (/ k l_m)) t_m) (* k k))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= 2e+275) {
		tmp = 2.0 / (((2.0 * ((t_m * t_m) / (l_m * l_m))) * t_m) * (k * k));
	} else {
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k));
	}
	return t_s * tmp;
}

Fortran

l_m =     private
t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))) <= 2d+275) then
        tmp = 2.0d0 / (((2.0d0 * ((t_m * t_m) / (l_m * l_m))) * t_m) * (k * k))
    else
        tmp = 2.0d0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if ((2.0 / ((((Math.pow(t_m, 3.0) / (l_m * l_m)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0))) <= 2e+275) {
		tmp = 2.0 / (((2.0 * ((t_m * t_m) / (l_m * l_m))) * t_m) * (k * k));
	} else {
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if (2.0 / ((((math.pow(t_m, 3.0) / (l_m * l_m)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))) <= 2e+275:
		tmp = 2.0 / (((2.0 * ((t_m * t_m) / (l_m * l_m))) * t_m) * (k * k))
	else:
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))) <= 2e+275)
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * Float64(Float64(t_m * t_m) / Float64(l_m * l_m))) * t_m) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * t_m) * Float64(k * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if ((2.0 / (((((t_m ^ 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0))) <= 2e+275)
		tmp = 2.0 / (((2.0 * ((t_m * t_m) / (l_m * l_m))) * t_m) * (k * k));
	else
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+275], N[(2.0 / N[(N[(N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 1.99999999999999992e275

   1. Initial program 80.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

   5. Applied rewrites 65.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(t \cdot \left({t}^{2} \cdot \left(\frac{1}{3} \cdot \frac{{k}^{2}}{{\ell}^{2}} + 2 \cdot \frac{1}{{\ell}^{2}}\right) + \frac{{k}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left({t}^{2} \cdot \left(\frac{1}{3} \cdot \frac{{k}^{2}}{{\ell}^{2}} + 2 \cdot \frac{1}{{\ell}^{2}}\right) + \frac{{k}^{2}}{{\ell}^{2}}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({t}^{2} \cdot \left(\frac{1}{3} \cdot \frac{{k}^{2}}{{\ell}^{2}} + 2 \cdot \frac{1}{{\ell}^{2}}\right) + \frac{{k}^{2}}{{\ell}^{2}}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

   8. Applied rewrites 67.5%

      \[\leadsto \frac{2}{\left(\left(\left(\frac{\left(k \cdot k\right) \cdot 0.3333333333333333}{\ell \cdot \ell} - -2 \cdot {\ell}^{-2}\right) \cdot \left(t \cdot t\right) + \frac{k \cdot k}{\ell \cdot \ell}\right) \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

       3. pow2 N/A

          \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{t \cdot t}{{\ell}^{2}}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{t \cdot t}{{\ell}^{2}}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

       5. pow2 N/A

          \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

       6. lift-*.f64 65.5

          \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

   11. Applied rewrites 65.5%

       \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

   if 1.99999999999999992e275 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 18.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

   5. Applied rewrites 40.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(t \cdot \left({t}^{2} \cdot \left(\frac{1}{3} \cdot \frac{{k}^{2}}{{\ell}^{2}} + 2 \cdot \frac{1}{{\ell}^{2}}\right) + \frac{{k}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left({t}^{2} \cdot \left(\frac{1}{3} \cdot \frac{{k}^{2}}{{\ell}^{2}} + 2 \cdot \frac{1}{{\ell}^{2}}\right) + \frac{{k}^{2}}{{\ell}^{2}}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({t}^{2} \cdot \left(\frac{1}{3} \cdot \frac{{k}^{2}}{{\ell}^{2}} + 2 \cdot \frac{1}{{\ell}^{2}}\right) + \frac{{k}^{2}}{{\ell}^{2}}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

   8. Applied rewrites 36.3%

      \[\leadsto \frac{2}{\left(\left(\left(\frac{\left(k \cdot k\right) \cdot 0.3333333333333333}{\ell \cdot \ell} - -2 \cdot {\ell}^{-2}\right) \cdot \left(t \cdot t\right) + \frac{k \cdot k}{\ell \cdot \ell}\right) \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot t\right) \cdot \left(k \cdot k\right)} \]
   10. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{{\ell}^{2}} \cdot t\right) \cdot \left(k \cdot k\right)} \]

       2. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot t\right) \cdot \left(k \cdot k\right)} \]

       3. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

       6. lower-/.f64 55.6

          \[\leadsto \frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

   11. Applied rewrites 55.6%

       \[\leadsto \frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{2}{\left(\left(2 \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 59.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{2}{\left(\left(2 \cdot \frac{t\_m \cdot t\_m}{l\_m \cdot l\_m}\right) \cdot t\_m\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t\_m}{l\_m \cdot l\_m}\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       2e+275)
    (/ 2.0 (* (* (* 2.0 (/ (* t_m t_m) (* l_m l_m))) t_m) (* k k)))
    (/ 2.0 (* (* (* k k) (/ t_m (* l_m l_m))) (* k k))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= 2e+275) {
		tmp = 2.0 / (((2.0 * ((t_m * t_m) / (l_m * l_m))) * t_m) * (k * k));
	} else {
		tmp = 2.0 / (((k * k) * (t_m / (l_m * l_m))) * (k * k));
	}
	return t_s * tmp;
}

Fortran

l_m =     private
t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))) <= 2d+275) then
        tmp = 2.0d0 / (((2.0d0 * ((t_m * t_m) / (l_m * l_m))) * t_m) * (k * k))
    else
        tmp = 2.0d0 / (((k * k) * (t_m / (l_m * l_m))) * (k * k))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if ((2.0 / ((((Math.pow(t_m, 3.0) / (l_m * l_m)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0))) <= 2e+275) {
		tmp = 2.0 / (((2.0 * ((t_m * t_m) / (l_m * l_m))) * t_m) * (k * k));
	} else {
		tmp = 2.0 / (((k * k) * (t_m / (l_m * l_m))) * (k * k));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if (2.0 / ((((math.pow(t_m, 3.0) / (l_m * l_m)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))) <= 2e+275:
		tmp = 2.0 / (((2.0 * ((t_m * t_m) / (l_m * l_m))) * t_m) * (k * k))
	else:
		tmp = 2.0 / (((k * k) * (t_m / (l_m * l_m))) * (k * k))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))) <= 2e+275)
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * Float64(Float64(t_m * t_m) / Float64(l_m * l_m))) * t_m) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(t_m / Float64(l_m * l_m))) * Float64(k * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if ((2.0 / (((((t_m ^ 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0))) <= 2e+275)
		tmp = 2.0 / (((2.0 * ((t_m * t_m) / (l_m * l_m))) * t_m) * (k * k));
	else
		tmp = 2.0 / (((k * k) * (t_m / (l_m * l_m))) * (k * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+275], N[(2.0 / N[(N[(N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 1.99999999999999992e275

   1. Initial program 80.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

   5. Applied rewrites 65.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(t \cdot \left({t}^{2} \cdot \left(\frac{1}{3} \cdot \frac{{k}^{2}}{{\ell}^{2}} + 2 \cdot \frac{1}{{\ell}^{2}}\right) + \frac{{k}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left({t}^{2} \cdot \left(\frac{1}{3} \cdot \frac{{k}^{2}}{{\ell}^{2}} + 2 \cdot \frac{1}{{\ell}^{2}}\right) + \frac{{k}^{2}}{{\ell}^{2}}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({t}^{2} \cdot \left(\frac{1}{3} \cdot \frac{{k}^{2}}{{\ell}^{2}} + 2 \cdot \frac{1}{{\ell}^{2}}\right) + \frac{{k}^{2}}{{\ell}^{2}}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

   8. Applied rewrites 67.5%

      \[\leadsto \frac{2}{\left(\left(\left(\frac{\left(k \cdot k\right) \cdot 0.3333333333333333}{\ell \cdot \ell} - -2 \cdot {\ell}^{-2}\right) \cdot \left(t \cdot t\right) + \frac{k \cdot k}{\ell \cdot \ell}\right) \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

       3. pow2 N/A

          \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{t \cdot t}{{\ell}^{2}}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{t \cdot t}{{\ell}^{2}}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

       5. pow2 N/A

          \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

       6. lift-*.f64 65.5

          \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

   11. Applied rewrites 65.5%

       \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

   if 1.99999999999999992e275 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 18.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

   5. Applied rewrites 40.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\color{blue}{k} \cdot k\right)} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)} \]

      7. lift-*.f64 45.0

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)} \]

   8. Applied rewrites 45.0%

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{2}{\left(\left(2 \cdot \frac{t \cdot t}{\ell \cdot \ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 59.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{\left(k \cdot k\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t\_m}{l\_m \cdot l\_m}\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       2e+275)
    (/ (* l_m l_m) (* (* k k) (* (* t_m t_m) t_m)))
    (/ 2.0 (* (* (* k k) (/ t_m (* l_m l_m))) (* k k))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= 2e+275) {
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	} else {
		tmp = 2.0 / (((k * k) * (t_m / (l_m * l_m))) * (k * k));
	}
	return t_s * tmp;
}

Fortran

l_m =     private
t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))) <= 2d+275) then
        tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m))
    else
        tmp = 2.0d0 / (((k * k) * (t_m / (l_m * l_m))) * (k * k))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if ((2.0 / ((((Math.pow(t_m, 3.0) / (l_m * l_m)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0))) <= 2e+275) {
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	} else {
		tmp = 2.0 / (((k * k) * (t_m / (l_m * l_m))) * (k * k));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if (2.0 / ((((math.pow(t_m, 3.0) / (l_m * l_m)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))) <= 2e+275:
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m))
	else:
		tmp = 2.0 / (((k * k) * (t_m / (l_m * l_m))) * (k * k))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))) <= 2e+275)
		tmp = Float64(Float64(l_m * l_m) / Float64(Float64(k * k) * Float64(Float64(t_m * t_m) * t_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(t_m / Float64(l_m * l_m))) * Float64(k * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if ((2.0 / (((((t_m ^ 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0))) <= 2e+275)
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	else
		tmp = 2.0 / (((k * k) * (t_m / (l_m * l_m))) * (k * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+275], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 1.99999999999999992e275

   1. Initial program 80.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 64.6

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 64.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. unpow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. lift-*.f64 64.5

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   7. Applied rewrites 64.5%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   if 1.99999999999999992e275 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 18.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

   5. Applied rewrites 40.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\color{blue}{k} \cdot k\right)} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)} \]

      7. lift-*.f64 45.0

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)} \]

   8. Applied rewrites 45.0%

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 72.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\right) \cdot k\right) \cdot \left(\left(1 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 7.8e-7)
    (/
     2.0
     (*
      (* (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (sin k)) k)
      (+ (+ 1.0 (* (/ k t_m) (/ k t_m))) 1.0)))
    (/
     2.0
     (* (* (/ k l_m) (/ k l_m)) (/ (* t_m (pow (sin k) 2.0)) (cos k)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 7.8e-7) {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * k) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	} else {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * pow(sin(k), 2.0)) / cos(k)));
	}
	return t_s * tmp;
}

Fortran

l_m =     private
t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 7.8d-7) then
        tmp = 2.0d0 / (((exp(((log(t_m) * 3.0d0) - (log(l_m) * 2.0d0))) * sin(k)) * k) * ((1.0d0 + ((k / t_m) * (k / t_m))) + 1.0d0))
    else
        tmp = 2.0d0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ** 2.0d0)) / cos(k)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 7.8e-7) {
		tmp = 2.0 / (((Math.exp(((Math.log(t_m) * 3.0) - (Math.log(l_m) * 2.0))) * Math.sin(k)) * k) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	} else {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(k)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 7.8e-7:
		tmp = 2.0 / (((math.exp(((math.log(t_m) * 3.0) - (math.log(l_m) * 2.0))) * math.sin(k)) * k) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0))
	else:
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * math.pow(math.sin(k), 2.0)) / math.cos(k)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 7.8e-7)
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * sin(k)) * k) * Float64(Float64(1.0 + Float64(Float64(k / t_m) * Float64(k / t_m))) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 7.8e-7)
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * k) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	else
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ^ 2.0)) / cos(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 7.8e-7], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * N[(N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.80000000000000049e-7

   1. Initial program 56.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t} \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-log.f64 19.1

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 19.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\color{blue}{\left(\frac{k}{t}\right)}}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right) + 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t}} \cdot \frac{k}{t}\right) + 1\right)} \]

      6. lift-/.f64 19.1

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \color{blue}{\frac{k}{t}}\right) + 1\right)} \]

   6. Applied rewrites 19.1%

      \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1\right)} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \color{blue}{k}\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 18.2%

      \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \color{blue}{k}\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

   if 7.80000000000000049e-7 < k

   1. Initial program 42.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

   5. Applied rewrites 54.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\color{blue}{k} \cdot k\right)} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)} \]

      7. lift-*.f64 54.4

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)} \]

   8. Applied rewrites 54.4%

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   10. Step-by-step derivation

       1. times-frac N/A

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

       3. pow2 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

       4. pow2 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

       5. times-frac N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\cos k}} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\cos k}} \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

       9. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\cos k}}} \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos \color{blue}{k}}} \]

       11. lower-pow.f64 N/A

           \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

       12. lift-sin.f64 N/A

           \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

       13. lower-cos.f64 78.6

           \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

   11. Applied rewrites 78.6%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 72.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 7.8e-7)
    (/
     2.0
     (*
      (* (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) k) (tan k))
      (+ (+ 1.0 (* (/ k t_m) (/ k t_m))) 1.0)))
    (/
     2.0
     (* (* (/ k l_m) (/ k l_m)) (/ (* t_m (pow (sin k) 2.0)) (cos k)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 7.8e-7) {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * k) * tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	} else {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * pow(sin(k), 2.0)) / cos(k)));
	}
	return t_s * tmp;
}

Fortran

l_m =     private
t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 7.8d-7) then
        tmp = 2.0d0 / (((exp(((log(t_m) * 3.0d0) - (log(l_m) * 2.0d0))) * k) * tan(k)) * ((1.0d0 + ((k / t_m) * (k / t_m))) + 1.0d0))
    else
        tmp = 2.0d0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ** 2.0d0)) / cos(k)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 7.8e-7) {
		tmp = 2.0 / (((Math.exp(((Math.log(t_m) * 3.0) - (Math.log(l_m) * 2.0))) * k) * Math.tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	} else {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(k)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 7.8e-7:
		tmp = 2.0 / (((math.exp(((math.log(t_m) * 3.0) - (math.log(l_m) * 2.0))) * k) * math.tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0))
	else:
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * math.pow(math.sin(k), 2.0)) / math.cos(k)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 7.8e-7)
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * k) * tan(k)) * Float64(Float64(1.0 + Float64(Float64(k / t_m) * Float64(k / t_m))) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 7.8e-7)
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * k) * tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	else
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ^ 2.0)) / cos(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 7.8e-7], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * k), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.80000000000000049e-7

   1. Initial program 56.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t} \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-log.f64 19.1

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 19.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\color{blue}{\left(\frac{k}{t}\right)}}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right) + 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t}} \cdot \frac{k}{t}\right) + 1\right)} \]

      6. lift-/.f64 19.1

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \color{blue}{\frac{k}{t}}\right) + 1\right)} \]

   6. Applied rewrites 19.1%

      \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1\right)} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \color{blue}{k}\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 18.7%

      \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \color{blue}{k}\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

   if 7.80000000000000049e-7 < k

   1. Initial program 42.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

   5. Applied rewrites 54.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\color{blue}{k} \cdot k\right)} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)} \]

      7. lift-*.f64 54.4

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)} \]

   8. Applied rewrites 54.4%

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   10. Step-by-step derivation

       1. times-frac N/A

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

       3. pow2 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

       4. pow2 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

       5. times-frac N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\cos k}} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\cos k}} \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

       9. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\cos k}}} \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos \color{blue}{k}}} \]

       11. lower-pow.f64 N/A

           \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

       12. lift-sin.f64 N/A

           \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

       13. lower-cos.f64 78.6

           \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

   11. Applied rewrites 78.6%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 67.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-158}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{{t\_m}^{3}}{l\_m}}{l\_m} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right) + 1\right)}\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t\_m - 2 \cdot \log l\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.1e-158)
    (/
     2.0
     (*
      (* (* (/ (/ (pow t_m 3.0) l_m) l_m) k) (tan k))
      (+ (+ 1.0 (* (/ k t_m) (/ k t_m))) 1.0)))
    (if (<= k 2.2e-6)
      (/ 2.0 (* 2.0 (* (* k k) (exp (- (* 3.0 (log t_m)) (* 2.0 (log l_m)))))))
      (/
       2.0
       (* (* (/ k l_m) (/ k l_m)) (/ (* t_m (pow (sin k) 2.0)) (cos k))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 2.1e-158) {
		tmp = 2.0 / (((((pow(t_m, 3.0) / l_m) / l_m) * k) * tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	} else if (k <= 2.2e-6) {
		tmp = 2.0 / (2.0 * ((k * k) * exp(((3.0 * log(t_m)) - (2.0 * log(l_m))))));
	} else {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * pow(sin(k), 2.0)) / cos(k)));
	}
	return t_s * tmp;
}

Fortran

l_m =     private
t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.1d-158) then
        tmp = 2.0d0 / ((((((t_m ** 3.0d0) / l_m) / l_m) * k) * tan(k)) * ((1.0d0 + ((k / t_m) * (k / t_m))) + 1.0d0))
    else if (k <= 2.2d-6) then
        tmp = 2.0d0 / (2.0d0 * ((k * k) * exp(((3.0d0 * log(t_m)) - (2.0d0 * log(l_m))))))
    else
        tmp = 2.0d0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ** 2.0d0)) / cos(k)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 2.1e-158) {
		tmp = 2.0 / (((((Math.pow(t_m, 3.0) / l_m) / l_m) * k) * Math.tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	} else if (k <= 2.2e-6) {
		tmp = 2.0 / (2.0 * ((k * k) * Math.exp(((3.0 * Math.log(t_m)) - (2.0 * Math.log(l_m))))));
	} else {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(k)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 2.1e-158:
		tmp = 2.0 / (((((math.pow(t_m, 3.0) / l_m) / l_m) * k) * math.tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0))
	elif k <= 2.2e-6:
		tmp = 2.0 / (2.0 * ((k * k) * math.exp(((3.0 * math.log(t_m)) - (2.0 * math.log(l_m))))))
	else:
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * math.pow(math.sin(k), 2.0)) / math.cos(k)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 2.1e-158)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64((t_m ^ 3.0) / l_m) / l_m) * k) * tan(k)) * Float64(Float64(1.0 + Float64(Float64(k / t_m) * Float64(k / t_m))) + 1.0)));
	elseif (k <= 2.2e-6)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(k * k) * exp(Float64(Float64(3.0 * log(t_m)) - Float64(2.0 * log(l_m)))))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 2.1e-158)
		tmp = 2.0 / ((((((t_m ^ 3.0) / l_m) / l_m) * k) * tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	elseif (k <= 2.2e-6)
		tmp = 2.0 / (2.0 * ((k * k) * exp(((3.0 * log(t_m)) - (2.0 * log(l_m))))));
	else
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((t_m * (sin(k) ^ 2.0)) / cos(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 2.1e-158], N[(2.0 / N[(N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * k), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.2e-6], N[(2.0 / N[(2.0 * N[(N[(k * k), $MachinePrecision] * N[Exp[N[(N[(3.0 * N[Log[t$95$m], $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.09999999999999991e-158

   1. Initial program 56.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t} \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-log.f64 19.2

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 19.2%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\color{blue}{\left(\frac{k}{t}\right)}}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right) + 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t}} \cdot \frac{k}{t}\right) + 1\right)} \]

      6. lift-/.f64 19.2

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \color{blue}{\frac{k}{t}}\right) + 1\right)} \]

   6. Applied rewrites 19.2%

      \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1\right)} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \log t - 2 \cdot \log \ell} \cdot \color{blue}{k}\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \log t - 2 \cdot \log \ell} \cdot \color{blue}{k}\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      3. exp-diff N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{3 \cdot \log t}}{e^{2 \cdot \log \ell}} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{e^{2 \cdot \log \ell}} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      5. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{e^{2 \cdot \log \ell}} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{e^{\log \ell \cdot 2}} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      7. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{{\ell}^{2}} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      9. associate-/r* N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      12. lift-/.f64 58.4

          \[\leadsto \frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

   9. Applied rewrites 58.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot k\right)} \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

   if 2.09999999999999991e-158 < k < 2.2000000000000001e-6

   1. Initial program 58.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t} \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-log.f64 18.5

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 18.5%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\color{blue}{\left(\frac{k}{t}\right)}}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right) + 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1\right)} \]

      4. times-frac N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{{t}^{2}}}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{k \cdot \frac{k}{{t}^{2}}}\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{k \cdot \frac{k}{{t}^{2}}}\right) + 1\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + k \cdot \color{blue}{\frac{k}{{t}^{2}}}\right) + 1\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + k \cdot \frac{k}{\color{blue}{t \cdot t}}\right) + 1\right)} \]

      10. lift-*.f64 18.5

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + k \cdot \frac{k}{\color{blue}{t \cdot t}}\right) + 1\right)} \]

   6. Applied rewrites 18.5%

      \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{k \cdot \frac{k}{t \cdot t}}\right) + 1\right)} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \left({k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{2 \cdot \left({k}^{2} \cdot \color{blue}{e^{3 \cdot \log t - 2 \cdot \log \ell}}\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{\color{blue}{3 \cdot \log t - 2 \cdot \log \ell}}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{\color{blue}{3 \cdot \log t - 2 \cdot \log \ell}}\right)} \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)} \]

      8. lift-log.f64 N/A

         \[\leadsto \frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)} \]

      10. lift-log.f64 18.5

          \[\leadsto \frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)} \]

   9. Applied rewrites 18.5%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)}} \]

   if 2.2000000000000001e-6 < k

   1. Initial program 42.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

   5. Applied rewrites 54.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\color{blue}{k} \cdot k\right)} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)} \]

      7. lift-*.f64 54.4

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)} \]

   8. Applied rewrites 54.4%

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   10. Step-by-step derivation

       1. times-frac N/A

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

       3. pow2 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

       4. pow2 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

       5. times-frac N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\cos k}} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\cos k}} \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

       9. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\cos k}}} \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos \color{blue}{k}}} \]

       11. lower-pow.f64 N/A

           \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

       12. lift-sin.f64 N/A

           \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

       13. lower-cos.f64 78.6

           \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]

   11. Applied rewrites 78.6%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-158}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)}\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 63.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-158}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{{t\_m}^{3}}{l\_m}}{l\_m} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right) + 1\right)}\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t\_m - 2 \cdot \log l\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(l\_m \cdot l\_m\right) \cdot \cos k}{{\sin k}^{2} \cdot t\_m} \cdot 2}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.1e-158)
    (/
     2.0
     (*
      (* (* (/ (/ (pow t_m 3.0) l_m) l_m) k) (tan k))
      (+ (+ 1.0 (* (/ k t_m) (/ k t_m))) 1.0)))
    (if (<= k 2.2e-6)
      (/ 2.0 (* 2.0 (* (* k k) (exp (- (* 3.0 (log t_m)) (* 2.0 (log l_m)))))))
      (/
       (* (/ (* (* l_m l_m) (cos k)) (* (pow (sin k) 2.0) t_m)) 2.0)
       (* k k))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 2.1e-158) {
		tmp = 2.0 / (((((pow(t_m, 3.0) / l_m) / l_m) * k) * tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	} else if (k <= 2.2e-6) {
		tmp = 2.0 / (2.0 * ((k * k) * exp(((3.0 * log(t_m)) - (2.0 * log(l_m))))));
	} else {
		tmp = ((((l_m * l_m) * cos(k)) / (pow(sin(k), 2.0) * t_m)) * 2.0) / (k * k);
	}
	return t_s * tmp;
}

Fortran

l_m =     private
t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.1d-158) then
        tmp = 2.0d0 / ((((((t_m ** 3.0d0) / l_m) / l_m) * k) * tan(k)) * ((1.0d0 + ((k / t_m) * (k / t_m))) + 1.0d0))
    else if (k <= 2.2d-6) then
        tmp = 2.0d0 / (2.0d0 * ((k * k) * exp(((3.0d0 * log(t_m)) - (2.0d0 * log(l_m))))))
    else
        tmp = ((((l_m * l_m) * cos(k)) / ((sin(k) ** 2.0d0) * t_m)) * 2.0d0) / (k * k)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 2.1e-158) {
		tmp = 2.0 / (((((Math.pow(t_m, 3.0) / l_m) / l_m) * k) * Math.tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	} else if (k <= 2.2e-6) {
		tmp = 2.0 / (2.0 * ((k * k) * Math.exp(((3.0 * Math.log(t_m)) - (2.0 * Math.log(l_m))))));
	} else {
		tmp = ((((l_m * l_m) * Math.cos(k)) / (Math.pow(Math.sin(k), 2.0) * t_m)) * 2.0) / (k * k);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 2.1e-158:
		tmp = 2.0 / (((((math.pow(t_m, 3.0) / l_m) / l_m) * k) * math.tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0))
	elif k <= 2.2e-6:
		tmp = 2.0 / (2.0 * ((k * k) * math.exp(((3.0 * math.log(t_m)) - (2.0 * math.log(l_m))))))
	else:
		tmp = ((((l_m * l_m) * math.cos(k)) / (math.pow(math.sin(k), 2.0) * t_m)) * 2.0) / (k * k)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 2.1e-158)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64((t_m ^ 3.0) / l_m) / l_m) * k) * tan(k)) * Float64(Float64(1.0 + Float64(Float64(k / t_m) * Float64(k / t_m))) + 1.0)));
	elseif (k <= 2.2e-6)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(k * k) * exp(Float64(Float64(3.0 * log(t_m)) - Float64(2.0 * log(l_m)))))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(l_m * l_m) * cos(k)) / Float64((sin(k) ^ 2.0) * t_m)) * 2.0) / Float64(k * k));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 2.1e-158)
		tmp = 2.0 / ((((((t_m ^ 3.0) / l_m) / l_m) * k) * tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	elseif (k <= 2.2e-6)
		tmp = 2.0 / (2.0 * ((k * k) * exp(((3.0 * log(t_m)) - (2.0 * log(l_m))))));
	else
		tmp = ((((l_m * l_m) * cos(k)) / ((sin(k) ^ 2.0) * t_m)) * 2.0) / (k * k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 2.1e-158], N[(2.0 / N[(N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * k), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.2e-6], N[(2.0 / N[(2.0 * N[(N[(k * k), $MachinePrecision] * N[Exp[N[(N[(3.0 * N[Log[t$95$m], $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.09999999999999991e-158

   1. Initial program 56.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t} \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-log.f64 19.2

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 19.2%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\color{blue}{\left(\frac{k}{t}\right)}}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right) + 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t}} \cdot \frac{k}{t}\right) + 1\right)} \]

      6. lift-/.f64 19.2

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \color{blue}{\frac{k}{t}}\right) + 1\right)} \]

   6. Applied rewrites 19.2%

      \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1\right)} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \log t - 2 \cdot \log \ell} \cdot \color{blue}{k}\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \log t - 2 \cdot \log \ell} \cdot \color{blue}{k}\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      3. exp-diff N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{3 \cdot \log t}}{e^{2 \cdot \log \ell}} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{e^{2 \cdot \log \ell}} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      5. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{e^{2 \cdot \log \ell}} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{e^{\log \ell \cdot 2}} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      7. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{{\ell}^{2}} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      9. associate-/r* N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      12. lift-/.f64 58.4

          \[\leadsto \frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

   9. Applied rewrites 58.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot k\right)} \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

   if 2.09999999999999991e-158 < k < 2.2000000000000001e-6

   1. Initial program 58.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t} \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-log.f64 18.5

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 18.5%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\color{blue}{\left(\frac{k}{t}\right)}}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right) + 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1\right)} \]

      4. times-frac N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{{t}^{2}}}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{k \cdot \frac{k}{{t}^{2}}}\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{k \cdot \frac{k}{{t}^{2}}}\right) + 1\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + k \cdot \color{blue}{\frac{k}{{t}^{2}}}\right) + 1\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + k \cdot \frac{k}{\color{blue}{t \cdot t}}\right) + 1\right)} \]

      10. lift-*.f64 18.5

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + k \cdot \frac{k}{\color{blue}{t \cdot t}}\right) + 1\right)} \]

   6. Applied rewrites 18.5%

      \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{k \cdot \frac{k}{t \cdot t}}\right) + 1\right)} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \left({k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{2 \cdot \left({k}^{2} \cdot \color{blue}{e^{3 \cdot \log t - 2 \cdot \log \ell}}\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{\color{blue}{3 \cdot \log t - 2 \cdot \log \ell}}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{\color{blue}{3 \cdot \log t - 2 \cdot \log \ell}}\right)} \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)} \]

      8. lift-log.f64 N/A

         \[\leadsto \frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)} \]

      10. lift-log.f64 18.5

          \[\leadsto \frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)} \]

   9. Applied rewrites 18.5%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)}} \]

   if 2.2000000000000001e-6 < k

   1. Initial program 42.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 41.2

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 41.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\frac{-4 \cdot \frac{{\ell}^{2} \cdot \left(t \cdot \cos k\right)}{{k}^{2} \cdot {\sin k}^{2}} + 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}}{{k}^{2}}} \]

   8. Applied rewrites 53.9%

      \[\leadsto \color{blue}{\frac{\left(\left(\ell \cdot \ell\right) \cdot \frac{\cos k \cdot t}{{\left(\sin k \cdot k\right)}^{2}}\right) \cdot -4 - -2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\sin k}^{2} \cdot t}}{k \cdot k}} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}}{\color{blue}{k} \cdot k} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}} \cdot 2}{k \cdot k} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2} \cdot t} \cdot 2}{k \cdot k} \]

       3. pow2 N/A

          \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\sin k}^{2} \cdot t} \cdot 2}{k \cdot k} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\sin k}^{2} \cdot t} \cdot 2}{k \cdot k} \]

       5. lift-cos.f64 N/A

          \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\sin k}^{2} \cdot t} \cdot 2}{k \cdot k} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\sin k}^{2} \cdot t} \cdot 2}{k \cdot k} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\sin k}^{2} \cdot t} \cdot 2}{k \cdot k} \]

       8. lift-sin.f64 N/A

          \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\sin k}^{2} \cdot t} \cdot 2}{k \cdot k} \]

       9. lift-pow.f64 N/A

          \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\sin k}^{2} \cdot t} \cdot 2}{k \cdot k} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\sin k}^{2} \cdot t} \cdot 2}{k \cdot k} \]

       11. lift-/.f64 63.6

           \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\sin k}^{2} \cdot t} \cdot 2}{k \cdot k} \]

   11. Applied rewrites 63.6%

       \[\leadsto \frac{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\sin k}^{2} \cdot t} \cdot 2}{\color{blue}{k} \cdot k} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 27: 63.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-158}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{{t\_m}^{3}}{l\_m}}{l\_m} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right) + 1\right)}\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t\_m - 2 \cdot \log l\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{l\_m \cdot l\_m}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t\_m}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.1e-158)
    (/
     2.0
     (*
      (* (* (/ (/ (pow t_m 3.0) l_m) l_m) k) (tan k))
      (+ (+ 1.0 (* (/ k t_m) (/ k t_m))) 1.0)))
    (if (<= k 2.2e-6)
      (/ 2.0 (* 2.0 (* (* k k) (exp (- (* 3.0 (log t_m)) (* 2.0 (log l_m)))))))
      (*
       (* (/ (* l_m l_m) (* k k)) (/ (cos k) (* (pow (sin k) 2.0) t_m)))
       2.0)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 2.1e-158) {
		tmp = 2.0 / (((((pow(t_m, 3.0) / l_m) / l_m) * k) * tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	} else if (k <= 2.2e-6) {
		tmp = 2.0 / (2.0 * ((k * k) * exp(((3.0 * log(t_m)) - (2.0 * log(l_m))))));
	} else {
		tmp = (((l_m * l_m) / (k * k)) * (cos(k) / (pow(sin(k), 2.0) * t_m))) * 2.0;
	}
	return t_s * tmp;
}

Fortran

l_m =     private
t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.1d-158) then
        tmp = 2.0d0 / ((((((t_m ** 3.0d0) / l_m) / l_m) * k) * tan(k)) * ((1.0d0 + ((k / t_m) * (k / t_m))) + 1.0d0))
    else if (k <= 2.2d-6) then
        tmp = 2.0d0 / (2.0d0 * ((k * k) * exp(((3.0d0 * log(t_m)) - (2.0d0 * log(l_m))))))
    else
        tmp = (((l_m * l_m) / (k * k)) * (cos(k) / ((sin(k) ** 2.0d0) * t_m))) * 2.0d0
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 2.1e-158) {
		tmp = 2.0 / (((((Math.pow(t_m, 3.0) / l_m) / l_m) * k) * Math.tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	} else if (k <= 2.2e-6) {
		tmp = 2.0 / (2.0 * ((k * k) * Math.exp(((3.0 * Math.log(t_m)) - (2.0 * Math.log(l_m))))));
	} else {
		tmp = (((l_m * l_m) / (k * k)) * (Math.cos(k) / (Math.pow(Math.sin(k), 2.0) * t_m))) * 2.0;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 2.1e-158:
		tmp = 2.0 / (((((math.pow(t_m, 3.0) / l_m) / l_m) * k) * math.tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0))
	elif k <= 2.2e-6:
		tmp = 2.0 / (2.0 * ((k * k) * math.exp(((3.0 * math.log(t_m)) - (2.0 * math.log(l_m))))))
	else:
		tmp = (((l_m * l_m) / (k * k)) * (math.cos(k) / (math.pow(math.sin(k), 2.0) * t_m))) * 2.0
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 2.1e-158)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64((t_m ^ 3.0) / l_m) / l_m) * k) * tan(k)) * Float64(Float64(1.0 + Float64(Float64(k / t_m) * Float64(k / t_m))) + 1.0)));
	elseif (k <= 2.2e-6)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(k * k) * exp(Float64(Float64(3.0 * log(t_m)) - Float64(2.0 * log(l_m)))))));
	else
		tmp = Float64(Float64(Float64(Float64(l_m * l_m) / Float64(k * k)) * Float64(cos(k) / Float64((sin(k) ^ 2.0) * t_m))) * 2.0);
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 2.1e-158)
		tmp = 2.0 / ((((((t_m ^ 3.0) / l_m) / l_m) * k) * tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	elseif (k <= 2.2e-6)
		tmp = 2.0 / (2.0 * ((k * k) * exp(((3.0 * log(t_m)) - (2.0 * log(l_m))))));
	else
		tmp = (((l_m * l_m) / (k * k)) * (cos(k) / ((sin(k) ^ 2.0) * t_m))) * 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 2.1e-158], N[(2.0 / N[(N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * k), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.2e-6], N[(2.0 / N[(2.0 * N[(N[(k * k), $MachinePrecision] * N[Exp[N[(N[(3.0 * N[Log[t$95$m], $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.09999999999999991e-158

   1. Initial program 56.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t} \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-log.f64 19.2

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 19.2%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\color{blue}{\left(\frac{k}{t}\right)}}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right) + 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t}} \cdot \frac{k}{t}\right) + 1\right)} \]

      6. lift-/.f64 19.2

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \color{blue}{\frac{k}{t}}\right) + 1\right)} \]

   6. Applied rewrites 19.2%

      \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1\right)} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \log t - 2 \cdot \log \ell} \cdot \color{blue}{k}\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \log t - 2 \cdot \log \ell} \cdot \color{blue}{k}\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      3. exp-diff N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{3 \cdot \log t}}{e^{2 \cdot \log \ell}} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{e^{2 \cdot \log \ell}} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      5. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{e^{2 \cdot \log \ell}} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{e^{\log \ell \cdot 2}} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      7. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{{\ell}^{2}} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      9. associate-/r* N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

      12. lift-/.f64 58.4

          \[\leadsto \frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

   9. Applied rewrites 58.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot k\right)} \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)} \]

   if 2.09999999999999991e-158 < k < 2.2000000000000001e-6

   1. Initial program 58.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t} \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-log.f64 18.5

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 18.5%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\color{blue}{\left(\frac{k}{t}\right)}}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right) + 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1\right)} \]

      4. times-frac N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{{t}^{2}}}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{k \cdot \frac{k}{{t}^{2}}}\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{k \cdot \frac{k}{{t}^{2}}}\right) + 1\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + k \cdot \color{blue}{\frac{k}{{t}^{2}}}\right) + 1\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + k \cdot \frac{k}{\color{blue}{t \cdot t}}\right) + 1\right)} \]

      10. lift-*.f64 18.5

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + k \cdot \frac{k}{\color{blue}{t \cdot t}}\right) + 1\right)} \]

   6. Applied rewrites 18.5%

      \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{k \cdot \frac{k}{t \cdot t}}\right) + 1\right)} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \left({k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{2 \cdot \left({k}^{2} \cdot \color{blue}{e^{3 \cdot \log t - 2 \cdot \log \ell}}\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{\color{blue}{3 \cdot \log t - 2 \cdot \log \ell}}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{\color{blue}{3 \cdot \log t - 2 \cdot \log \ell}}\right)} \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)} \]

      8. lift-log.f64 N/A

         \[\leadsto \frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)} \]

      10. lift-log.f64 18.5

          \[\leadsto \frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)} \]

   9. Applied rewrites 18.5%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)}} \]

   if 2.2000000000000001e-6 < k

   1. Initial program 42.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{2} \]

   5. Applied rewrites 61.9%

      \[\leadsto \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-158}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t} \cdot \frac{k}{t}\right) + 1\right)}\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(k \cdot k\right) \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 71.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.5 \cdot 10^{-85}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot t\_m\right) \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;t\_m \leq 7.6 \cdot 10^{+54}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k \cdot k}{t\_m \cdot t\_m}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\_m\right)}^{2} \cdot \left(\left(k \cdot k\right) \cdot 0.3333333333333333 + 2\right) + {k}^{4}}{l\_m \cdot l\_m} \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.5e-85)
    (/ 2.0 (* (* (* (/ k l_m) (/ k l_m)) t_m) (* k k)))
    (if (<= t_m 7.6e+54)
      (/
       2.0
       (*
        (* (* (/ (* (* t_m t_m) t_m) (* l_m l_m)) (sin k)) (tan k))
        (+ (+ 1.0 (/ (* k k) (* t_m t_m))) 1.0)))
      (/
       2.0
       (*
        (/
         (+
          (* (pow (* k t_m) 2.0) (+ (* (* k k) 0.3333333333333333) 2.0))
          (pow k 4.0))
         (* l_m l_m))
        t_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 7.5e-85) {
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k));
	} else if (t_m <= 7.6e+54) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + ((k * k) / (t_m * t_m))) + 1.0));
	} else {
		tmp = 2.0 / ((((pow((k * t_m), 2.0) * (((k * k) * 0.3333333333333333) + 2.0)) + pow(k, 4.0)) / (l_m * l_m)) * t_m);
	}
	return t_s * tmp;
}

Fortran

l_m =     private
t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 7.5d-85) then
        tmp = 2.0d0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k))
    else if (t_m <= 7.6d+54) then
        tmp = 2.0d0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0d0 + ((k * k) / (t_m * t_m))) + 1.0d0))
    else
        tmp = 2.0d0 / ((((((k * t_m) ** 2.0d0) * (((k * k) * 0.3333333333333333d0) + 2.0d0)) + (k ** 4.0d0)) / (l_m * l_m)) * t_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 7.5e-85) {
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k));
	} else if (t_m <= 7.6e+54) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + ((k * k) / (t_m * t_m))) + 1.0));
	} else {
		tmp = 2.0 / ((((Math.pow((k * t_m), 2.0) * (((k * k) * 0.3333333333333333) + 2.0)) + Math.pow(k, 4.0)) / (l_m * l_m)) * t_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 7.5e-85:
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k))
	elif t_m <= 7.6e+54:
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * math.sin(k)) * math.tan(k)) * ((1.0 + ((k * k) / (t_m * t_m))) + 1.0))
	else:
		tmp = 2.0 / ((((math.pow((k * t_m), 2.0) * (((k * k) * 0.3333333333333333) + 2.0)) + math.pow(k, 4.0)) / (l_m * l_m)) * t_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 7.5e-85)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * t_m) * Float64(k * k)));
	elseif (t_m <= 7.6e+54)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + Float64(Float64(k * k) / Float64(t_m * t_m))) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * Float64(Float64(Float64(k * k) * 0.3333333333333333) + 2.0)) + (k ^ 4.0)) / Float64(l_m * l_m)) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 7.5e-85)
		tmp = 2.0 / ((((k / l_m) * (k / l_m)) * t_m) * (k * k));
	elseif (t_m <= 7.6e+54)
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + ((k * k) / (t_m * t_m))) + 1.0));
	else
		tmp = 2.0 / ((((((k * t_m) ^ 2.0) * (((k * k) * 0.3333333333333333) + 2.0)) + (k ^ 4.0)) / (l_m * l_m)) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.5e-85], N[(2.0 / N[(N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.6e+54], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(k * k), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 7.5000000000000003e-85

   1. Initial program 46.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

   5. Applied rewrites 55.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(t \cdot \left({t}^{2} \cdot \left(\frac{1}{3} \cdot \frac{{k}^{2}}{{\ell}^{2}} + 2 \cdot \frac{1}{{\ell}^{2}}\right) + \frac{{k}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left({t}^{2} \cdot \left(\frac{1}{3} \cdot \frac{{k}^{2}}{{\ell}^{2}} + 2 \cdot \frac{1}{{\ell}^{2}}\right) + \frac{{k}^{2}}{{\ell}^{2}}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({t}^{2} \cdot \left(\frac{1}{3} \cdot \frac{{k}^{2}}{{\ell}^{2}} + 2 \cdot \frac{1}{{\ell}^{2}}\right) + \frac{{k}^{2}}{{\ell}^{2}}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

   8. Applied rewrites 51.8%

      \[\leadsto \frac{2}{\left(\left(\left(\frac{\left(k \cdot k\right) \cdot 0.3333333333333333}{\ell \cdot \ell} - -2 \cdot {\ell}^{-2}\right) \cdot \left(t \cdot t\right) + \frac{k \cdot k}{\ell \cdot \ell}\right) \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot t\right) \cdot \left(k \cdot k\right)} \]
   10. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{{\ell}^{2}} \cdot t\right) \cdot \left(k \cdot k\right)} \]

       2. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot t\right) \cdot \left(k \cdot k\right)} \]

       3. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

       6. lower-/.f64 59.6

          \[\leadsto \frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

   11. Applied rewrites 59.6%

       \[\leadsto \frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

   if 7.5000000000000003e-85 < t < 7.6000000000000005e54

   1. Initial program 75.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\color{blue}{\left(\frac{k}{t}\right)}}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right) + 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1\right)} \]

      4. frac-times N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) + 1\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{\color{blue}{{k}^{2}}}{t \cdot t}\right) + 1\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{{k}^{2}}{\color{blue}{{t}^{2}}}\right) + 1\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{{k}^{2}}{{t}^{2}}}\right) + 1\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{\color{blue}{k \cdot k}}{{t}^{2}}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{\color{blue}{k \cdot k}}{{t}^{2}}\right) + 1\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{t \cdot t}}\right) + 1\right)} \]

      11. lower-*.f64 75.8

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{t \cdot t}}\right) + 1\right)} \]

   4. Applied rewrites 75.8%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k \cdot k}{t \cdot t}\right) + 1\right)} \]

      2. unpow3 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k \cdot k}{t \cdot t}\right) + 1\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{2}} \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k \cdot k}{t \cdot t}\right) + 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{2} \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k \cdot k}{t \cdot t}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right)} \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k \cdot k}{t \cdot t}\right) + 1\right)} \]

      6. lift-*.f64 75.7

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right)} \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k \cdot k}{t \cdot t}\right) + 1\right)} \]

   6. Applied rewrites 75.7%

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k \cdot k}{t \cdot t}\right) + 1\right)} \]

   if 7.6000000000000005e54 < t

   1. Initial program 62.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

   5. Applied rewrites 53.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(0.3333333333333333 \cdot {t}^{3} + t\right) \cdot \left(k \cdot k\right) + 2 \cdot {t}^{3}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{t \cdot \color{blue}{\left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(2 + \frac{1}{3} \cdot {k}^{2}\right)\right)}{{\ell}^{2}} + \frac{{k}^{4}}{{\ell}^{2}}\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(2 + \frac{1}{3} \cdot {k}^{2}\right)\right)}{{\ell}^{2}} + \frac{{k}^{4}}{{\ell}^{2}}\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(2 + \frac{1}{3} \cdot {k}^{2}\right)\right)}{{\ell}^{2}} + \frac{{k}^{4}}{{\ell}^{2}}\right) \cdot t} \]

   8. Applied rewrites 73.3%

      \[\leadsto \frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot \left(\left(k \cdot k\right) \cdot 0.3333333333333333 + 2\right) + {k}^{4}}{\ell \cdot \ell} \cdot \color{blue}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-85}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{+54}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k \cdot k}{t \cdot t}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot \left(\left(k \cdot k\right) \cdot 0.3333333333333333 + 2\right) + {k}^{4}}{\ell \cdot \ell} \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 50.9% accurate, 12.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{l\_m \cdot l\_m}{\left(k \cdot k\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (* t_s (/ (* l_m l_m) (* (* k k) (* (* t_m t_m) t_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	return t_s * ((l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m)));
}

Fortran

l_m =     private
t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = t_s * ((l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m)))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	return t_s * ((l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m)));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	return t_s * ((l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m)))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	return Float64(t_s * Float64(Float64(l_m * l_m) / Float64(Float64(k * k) * Float64(Float64(t_m * t_m) * t_m))))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k)
	tmp = t_s * ((l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m)));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   7. lift-pow.f64 47.7

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

5. Applied rewrites 47.7%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   2. unpow3 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   3. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

   5. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   6. lift-*.f64 47.7

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

7. Applied rewrites 47.7%

   \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

8. Final simplification 47.7%

   \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025058 
(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))))