Toniolo and Linder, Equation (10+)

Percentage Accurate: 53.9% → 84.7%

Time: 8.8s

Alternatives: 9

Speedup: N/A×

• 26.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.9% 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: 84.7% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{\sin k}^{2}}{\cos k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.5 \cdot 10^{-251}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{t\_m}{l\_m}}{l\_m} \cdot t\_2\right) \cdot k\right) \cdot k}\\ \mathbf{elif}\;t\_m \leq 3.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\left(t\_2 \cdot \left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right)\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \end{array} \]

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 (/ (pow (sin k) 2.0) (cos k))))
   (*
    t_s
    (if (<= t_m 5.5e-251)
      (/ 2.0 (* (* (* (/ (/ t_m l_m) l_m) t_2) k) k))
      (if (<= t_m 3.4e-7)
        (/ 2.0 (* (* t_2 (* (/ k l_m) (/ k l_m))) t_m))
        (/
         2.0
         (*
          (* (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (sin k)) (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 t_2 = pow(sin(k), 2.0) / cos(k);
	double tmp;
	if (t_m <= 5.5e-251) {
		tmp = 2.0 / (((((t_m / l_m) / l_m) * t_2) * k) * k);
	} else if (t_m <= 3.4e-7) {
		tmp = 2.0 / ((t_2 * ((k / l_m) * (k / l_m))) * t_m);
	} else {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * 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) :: t_2
    real(8) :: tmp
    t_2 = (sin(k) ** 2.0d0) / cos(k)
    if (t_m <= 5.5d-251) then
        tmp = 2.0d0 / (((((t_m / l_m) / l_m) * t_2) * k) * k)
    else if (t_m <= 3.4d-7) then
        tmp = 2.0d0 / ((t_2 * ((k / l_m) * (k / l_m))) * t_m)
    else
        tmp = 2.0d0 / (((exp(((log(t_m) * 3.0d0) - (log(l_m) * 2.0d0))) * sin(k)) * 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 t_2 = Math.pow(Math.sin(k), 2.0) / Math.cos(k);
	double tmp;
	if (t_m <= 5.5e-251) {
		tmp = 2.0 / (((((t_m / l_m) / l_m) * t_2) * k) * k);
	} else if (t_m <= 3.4e-7) {
		tmp = 2.0 / ((t_2 * ((k / l_m) * (k / l_m))) * t_m);
	} 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 + 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):
	t_2 = math.pow(math.sin(k), 2.0) / math.cos(k)
	tmp = 0
	if t_m <= 5.5e-251:
		tmp = 2.0 / (((((t_m / l_m) / l_m) * t_2) * k) * k)
	elif t_m <= 3.4e-7:
		tmp = 2.0 / ((t_2 * ((k / l_m) * (k / l_m))) * t_m)
	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 + 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)
	t_2 = Float64((sin(k) ^ 2.0) / cos(k))
	tmp = 0.0
	if (t_m <= 5.5e-251)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m / l_m) / l_m) * t_2) * k) * k));
	elseif (t_m <= 3.4e-7)
		tmp = Float64(2.0 / Float64(Float64(t_2 * Float64(Float64(k / l_m) * Float64(k / l_m))) * t_m));
	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 / 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)
	t_2 = (sin(k) ^ 2.0) / cos(k);
	tmp = 0.0;
	if (t_m <= 5.5e-251)
		tmp = 2.0 / (((((t_m / l_m) / l_m) * t_2) * k) * k);
	elseif (t_m <= 3.4e-7)
		tmp = 2.0 / ((t_2 * ((k / l_m) * (k / l_m))) * t_m);
	else
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * 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_] := Block[{t$95$2 = N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.5e-251], N[(2.0 / N[(N[(N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.4e-7], N[(2.0 / N[(N[(t$95$2 * N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $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[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 5.5e-251

   1. Initial program 54.1%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 61.8

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

   5. Applied rewrites 61.8%

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

   7. Applied rewrites 68.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 73.2

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

   9. Applied rewrites 73.2%

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

   if 5.5e-251 < t < 3.39999999999999974e-7

   1. Initial program 45.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 72.9%

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

   6. Taylor expanded in t around 0

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

      1. frac-times N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-/.f64 73.0

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. times-frac N/A

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

   8. Applied rewrites 91.9%

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

   if 3.39999999999999974e-7 < t

   1. Initial program 65.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 33.0

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

   4. Applied rewrites 33.0%

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

Alternative 2: 82.1% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{\sin k}^{2}}{\cos k}\\ t_3 := {\left(t\_m \cdot \sin k\right)}^{1}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.4 \cdot 10^{-140}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{t\_m}{l\_m}}{l\_m} \cdot t\_2\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{\frac{k}{l\_m} \cdot k}{l\_m}, t\_2, \frac{\mathsf{fma}\left(t\_3, t\_3, t\_3 \cdot t\_3\right)}{\cos k \cdot \left(l\_m \cdot l\_m\right)}\right) \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

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 (sin k) 2.0) (cos k))) (t_3 (pow (* t_m (sin k)) 1.0)))
   (*
    t_s
    (if (<= t_m 4.4e-140)
      (/ 2.0 (* (* (* (/ (/ t_m l_m) l_m) t_2) k) k))
      (/
       2.0
       (*
        (fma
         (/ (* (/ k l_m) k) l_m)
         t_2
         (/ (fma t_3 t_3 (* t_3 t_3)) (* (cos k) (* 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 t_2 = pow(sin(k), 2.0) / cos(k);
	double t_3 = pow((t_m * sin(k)), 1.0);
	double tmp;
	if (t_m <= 4.4e-140) {
		tmp = 2.0 / (((((t_m / l_m) / l_m) * t_2) * k) * k);
	} else {
		tmp = 2.0 / (fma((((k / l_m) * k) / l_m), t_2, (fma(t_3, t_3, (t_3 * t_3)) / (cos(k) * (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)
	t_2 = Float64((sin(k) ^ 2.0) / cos(k))
	t_3 = Float64(t_m * sin(k)) ^ 1.0
	tmp = 0.0
	if (t_m <= 4.4e-140)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m / l_m) / l_m) * t_2) * k) * k));
	else
		tmp = Float64(2.0 / Float64(fma(Float64(Float64(Float64(k / l_m) * k) / l_m), t_2, Float64(fma(t_3, t_3, Float64(t_3 * t_3)) / Float64(cos(k) * Float64(l_m * l_m)))) * t_m));
	end
	return Float64(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[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision], 1.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.4e-140], N[(2.0 / N[(N[(N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k / l$95$m), $MachinePrecision] * k), $MachinePrecision] / l$95$m), $MachinePrecision] * t$95$2 + N[(N[(t$95$3 * t$95$3 + N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 4.3999999999999998e-140

   1. Initial program 50.1%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 61.0

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

   5. Applied rewrites 61.0%

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

   7. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.3

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

   9. Applied rewrites 74.3%

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

   if 4.3999999999999998e-140 < t

   1. Initial program 61.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 82.9

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

   7. Applied rewrites 82.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 89.1

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

   9. Applied rewrites 89.1%

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

Alternative 3: 81.5% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k}^{2}\\ t_3 := \cos k \cdot \left(l\_m \cdot l\_m\right)\\ t_4 := \frac{t\_2}{\cos k}\\ t_5 := {\left(t\_m \cdot \sin k\right)}^{1}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.4 \cdot 10^{-140}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{t\_m}{l\_m}}{l\_m} \cdot t\_4\right) \cdot k\right) \cdot k}\\ \mathbf{elif}\;t\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{\frac{k}{l\_m} \cdot k}{l\_m}, t\_4, \frac{\mathsf{fma}\left(t\_5, t\_5, t\_2 \cdot \left(t\_m \cdot t\_m\right)\right)}{t\_3}\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k \cdot k}{l\_m \cdot l\_m}, t\_4, \frac{\mathsf{fma}\left(t\_5, t\_5, t\_5 \cdot t\_5\right)}{t\_3}\right) \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

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 (sin k) 2.0))
        (t_3 (* (cos k) (* l_m l_m)))
        (t_4 (/ t_2 (cos k)))
        (t_5 (pow (* t_m (sin k)) 1.0)))
   (*
    t_s
    (if (<= t_m 4.4e-140)
      (/ 2.0 (* (* (* (/ (/ t_m l_m) l_m) t_4) k) k))
      (if (<= t_m 1.35e+154)
        (/
         2.0
         (*
          (fma
           (/ (* (/ k l_m) k) l_m)
           t_4
           (/ (fma t_5 t_5 (* t_2 (* t_m t_m))) t_3))
          t_m))
        (/
         2.0
         (*
          (fma (/ (* k k) (* l_m l_m)) t_4 (/ (fma t_5 t_5 (* t_5 t_5)) t_3))
          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 t_2 = pow(sin(k), 2.0);
	double t_3 = cos(k) * (l_m * l_m);
	double t_4 = t_2 / cos(k);
	double t_5 = pow((t_m * sin(k)), 1.0);
	double tmp;
	if (t_m <= 4.4e-140) {
		tmp = 2.0 / (((((t_m / l_m) / l_m) * t_4) * k) * k);
	} else if (t_m <= 1.35e+154) {
		tmp = 2.0 / (fma((((k / l_m) * k) / l_m), t_4, (fma(t_5, t_5, (t_2 * (t_m * t_m))) / t_3)) * t_m);
	} else {
		tmp = 2.0 / (fma(((k * k) / (l_m * l_m)), t_4, (fma(t_5, t_5, (t_5 * t_5)) / t_3)) * 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)
	t_2 = sin(k) ^ 2.0
	t_3 = Float64(cos(k) * Float64(l_m * l_m))
	t_4 = Float64(t_2 / cos(k))
	t_5 = Float64(t_m * sin(k)) ^ 1.0
	tmp = 0.0
	if (t_m <= 4.4e-140)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m / l_m) / l_m) * t_4) * k) * k));
	elseif (t_m <= 1.35e+154)
		tmp = Float64(2.0 / Float64(fma(Float64(Float64(Float64(k / l_m) * k) / l_m), t_4, Float64(fma(t_5, t_5, Float64(t_2 * Float64(t_m * t_m))) / t_3)) * t_m));
	else
		tmp = Float64(2.0 / Float64(fma(Float64(Float64(k * k) / Float64(l_m * l_m)), t_4, Float64(fma(t_5, t_5, Float64(t_5 * t_5)) / t_3)) * t_m));
	end
	return Float64(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[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 / N[Cos[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision], 1.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.4e-140], N[(2.0 / N[(N[(N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * t$95$4), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.35e+154], N[(2.0 / N[(N[(N[(N[(N[(k / l$95$m), $MachinePrecision] * k), $MachinePrecision] / l$95$m), $MachinePrecision] * t$95$4 + N[(N[(t$95$5 * t$95$5 + N[(t$95$2 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$4 + N[(N[(t$95$5 * t$95$5 + N[(t$95$5 * t$95$5), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 4.3999999999999998e-140

   1. Initial program 50.1%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 61.0

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

   5. Applied rewrites 61.0%

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

   7. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.3

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

   9. Applied rewrites 74.3%

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

   if 4.3999999999999998e-140 < t < 1.35000000000000003e154

   1. Initial program 66.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 84.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 87.6

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

   7. Applied rewrites 87.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 96.5

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

   9. Applied rewrites 96.5%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\frac{k}{\ell} \cdot k}{\ell}, \frac{{\sin k}^{2}}{\cos k}, \frac{\mathsf{fma}\left({\left(t \cdot \sin k\right)}^{1}, {\left(t \cdot \sin k\right)}^{1}, {\left(t \cdot \sin k\right)}^{1} \cdot {\left(t \cdot \sin k\right)}^{1}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}\right) \cdot t} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-sin.f64 N/A

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

       5. lift-pow.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-sin.f64 N/A

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

       8. pow-prod-up N/A

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

       9. metadata-eval N/A

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

       10. pow-prod-down N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. lift-sin.f64 N/A

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

       14. lift-pow.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 94.2

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

   11. Applied rewrites 94.2%

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

   if 1.35000000000000003e154 < t

   1. Initial program 51.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 61.1%

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

Alternative 4: 81.1% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{\sin k}^{2}}{\cos k}\\ t_3 := {\left(t\_m \cdot \sin k\right)}^{1}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 10^{+265}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k \cdot k}{l\_m \cdot l\_m}, t\_2, \frac{\mathsf{fma}\left(t\_3, t\_3, t\_3 \cdot t\_3\right)}{\cos k \cdot \left(l\_m \cdot l\_m\right)}\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_2 \cdot \left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right)\right) \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

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 (sin k) 2.0) (cos k))) (t_3 (pow (* t_m (sin k)) 1.0)))
   (*
    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)))
         1e+265)
      (/
       2.0
       (*
        (fma
         (/ (* k k) (* l_m l_m))
         t_2
         (/ (fma t_3 t_3 (* t_3 t_3)) (* (cos k) (* l_m l_m))))
        t_m))
      (/ 2.0 (* (* t_2 (* (/ k l_m) (/ k 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 t_2 = pow(sin(k), 2.0) / cos(k);
	double t_3 = pow((t_m * sin(k)), 1.0);
	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))) <= 1e+265) {
		tmp = 2.0 / (fma(((k * k) / (l_m * l_m)), t_2, (fma(t_3, t_3, (t_3 * t_3)) / (cos(k) * (l_m * l_m)))) * t_m);
	} else {
		tmp = 2.0 / ((t_2 * ((k / l_m) * (k / 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)
	t_2 = Float64((sin(k) ^ 2.0) / cos(k))
	t_3 = Float64(t_m * sin(k)) ^ 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)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))) <= 1e+265)
		tmp = Float64(2.0 / Float64(fma(Float64(Float64(k * k) / Float64(l_m * l_m)), t_2, Float64(fma(t_3, t_3, Float64(t_3 * t_3)) / Float64(cos(k) * Float64(l_m * l_m)))) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(t_2 * Float64(Float64(k / l_m) * Float64(k / l_m))) * t_m));
	end
	return Float64(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[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(t$95$m * N[Sin[k], $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] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+265], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$2 + N[(N[(t$95$3 * t$95$3 + N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$2 * N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $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)))) < 1.00000000000000007e265

   1. Initial program 83.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.4%

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

   if 1.00000000000000007e265 < (/.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 17.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in t around 0

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

      1. frac-times N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-/.f64 54.4

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. times-frac N/A

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

   8. Applied rewrites 74.4%

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

Alternative 5: 79.8% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{\sin k}^{2}}{\cos k}\\ t_3 := {\left(t\_m \cdot \sin k\right)}^{1}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 10^{+265}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k \cdot k}{l\_m \cdot l\_m}, t\_2, \frac{\mathsf{fma}\left(t\_3, t\_3, e^{\log \left({\left(\sin k \cdot t\_m\right)}^{2}\right)}\right)}{\cos k \cdot \left(l\_m \cdot l\_m\right)}\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_2 \cdot \left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right)\right) \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

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 (sin k) 2.0) (cos k))) (t_3 (pow (* t_m (sin k)) 1.0)))
   (*
    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)))
         1e+265)
      (/
       2.0
       (*
        (fma
         (/ (* k k) (* l_m l_m))
         t_2
         (/
          (fma t_3 t_3 (exp (log (pow (* (sin k) t_m) 2.0))))
          (* (cos k) (* l_m l_m))))
        t_m))
      (/ 2.0 (* (* t_2 (* (/ k l_m) (/ k 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 t_2 = pow(sin(k), 2.0) / cos(k);
	double t_3 = pow((t_m * sin(k)), 1.0);
	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))) <= 1e+265) {
		tmp = 2.0 / (fma(((k * k) / (l_m * l_m)), t_2, (fma(t_3, t_3, exp(log(pow((sin(k) * t_m), 2.0)))) / (cos(k) * (l_m * l_m)))) * t_m);
	} else {
		tmp = 2.0 / ((t_2 * ((k / l_m) * (k / 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)
	t_2 = Float64((sin(k) ^ 2.0) / cos(k))
	t_3 = Float64(t_m * sin(k)) ^ 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)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))) <= 1e+265)
		tmp = Float64(2.0 / Float64(fma(Float64(Float64(k * k) / Float64(l_m * l_m)), t_2, Float64(fma(t_3, t_3, exp(log((Float64(sin(k) * t_m) ^ 2.0)))) / Float64(cos(k) * Float64(l_m * l_m)))) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(t_2 * Float64(Float64(k / l_m) * Float64(k / l_m))) * t_m));
	end
	return Float64(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[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(t$95$m * N[Sin[k], $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] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+265], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$2 + N[(N[(t$95$3 * t$95$3 + N[Exp[N[Log[N[Power[N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$2 * N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $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)))) < 1.00000000000000007e265

   1. Initial program 83.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow-prod-down N/A

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

      9. unpow1 N/A

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

      10. unpow1 N/A

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

      11. pow-to-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 86.7%

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

   if 1.00000000000000007e265 < (/.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 17.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in t around 0

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

      1. frac-times N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-/.f64 54.4

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. times-frac N/A

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

   8. Applied rewrites 74.4%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 10^{+265}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k \cdot k}{\ell \cdot \ell}, \frac{{\sin k}^{2}}{\cos k}, \frac{\mathsf{fma}\left({\left(t \cdot \sin k\right)}^{1}, {\left(t \cdot \sin k\right)}^{1}, e^{\log \left({\left(\sin k \cdot t\right)}^{2}\right)}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.8% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{\sin k}^{2}}{\cos k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 6 \cdot 10^{+61}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{t\_m}{l\_m}}{l\_m} \cdot t\_2\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_2 \cdot \left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right)\right) \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

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 (sin k) 2.0) (cos k))))
   (*
    t_s
    (if (<= l_m 6e+61)
      (/ 2.0 (* (* (* (/ (/ t_m l_m) l_m) t_2) k) k))
      (/ 2.0 (* (* t_2 (* (/ k l_m) (/ k 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 t_2 = pow(sin(k), 2.0) / cos(k);
	double tmp;
	if (l_m <= 6e+61) {
		tmp = 2.0 / (((((t_m / l_m) / l_m) * t_2) * k) * k);
	} else {
		tmp = 2.0 / ((t_2 * ((k / l_m) * (k / 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) :: t_2
    real(8) :: tmp
    t_2 = (sin(k) ** 2.0d0) / cos(k)
    if (l_m <= 6d+61) then
        tmp = 2.0d0 / (((((t_m / l_m) / l_m) * t_2) * k) * k)
    else
        tmp = 2.0d0 / ((t_2 * ((k / l_m) * (k / 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 t_2 = Math.pow(Math.sin(k), 2.0) / Math.cos(k);
	double tmp;
	if (l_m <= 6e+61) {
		tmp = 2.0 / (((((t_m / l_m) / l_m) * t_2) * k) * k);
	} else {
		tmp = 2.0 / ((t_2 * ((k / l_m) * (k / 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):
	t_2 = math.pow(math.sin(k), 2.0) / math.cos(k)
	tmp = 0
	if l_m <= 6e+61:
		tmp = 2.0 / (((((t_m / l_m) / l_m) * t_2) * k) * k)
	else:
		tmp = 2.0 / ((t_2 * ((k / l_m) * (k / 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)
	t_2 = Float64((sin(k) ^ 2.0) / cos(k))
	tmp = 0.0
	if (l_m <= 6e+61)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m / l_m) / l_m) * t_2) * k) * k));
	else
		tmp = Float64(2.0 / Float64(Float64(t_2 * Float64(Float64(k / l_m) * Float64(k / 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)
	t_2 = (sin(k) ^ 2.0) / cos(k);
	tmp = 0.0;
	if (l_m <= 6e+61)
		tmp = 2.0 / (((((t_m / l_m) / l_m) * t_2) * k) * k);
	else
		tmp = 2.0 / ((t_2 * ((k / l_m) * (k / 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_] := Block[{t$95$2 = N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l$95$m, 6e+61], N[(2.0 / N[(N[(N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$2 * N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6e61

   1. Initial program 55.4%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 60.4

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

   5. Applied rewrites 60.4%

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

   7. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 72.5

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

   9. Applied rewrites 72.5%

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

   if 6e61 < l

   1. Initial program 49.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in t around 0

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

      1. frac-times N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-/.f64 66.5

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. times-frac N/A

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

   8. Applied rewrites 75.3%

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

Alternative 7: 72.0% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{\sin k}^{2}}{\cos k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 4 \cdot 10^{+123}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{l\_m \cdot l\_m} \cdot t\_2\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_2 \cdot \left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right)\right) \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

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 (sin k) 2.0) (cos k))))
   (*
    t_s
    (if (<= (* l_m l_m) 4e+123)
      (/ 2.0 (* (* (* (/ t_m (* l_m l_m)) t_2) k) k))
      (/ 2.0 (* (* t_2 (* (/ k l_m) (/ k 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 t_2 = pow(sin(k), 2.0) / cos(k);
	double tmp;
	if ((l_m * l_m) <= 4e+123) {
		tmp = 2.0 / ((((t_m / (l_m * l_m)) * t_2) * k) * k);
	} else {
		tmp = 2.0 / ((t_2 * ((k / l_m) * (k / 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) :: t_2
    real(8) :: tmp
    t_2 = (sin(k) ** 2.0d0) / cos(k)
    if ((l_m * l_m) <= 4d+123) then
        tmp = 2.0d0 / ((((t_m / (l_m * l_m)) * t_2) * k) * k)
    else
        tmp = 2.0d0 / ((t_2 * ((k / l_m) * (k / 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 t_2 = Math.pow(Math.sin(k), 2.0) / Math.cos(k);
	double tmp;
	if ((l_m * l_m) <= 4e+123) {
		tmp = 2.0 / ((((t_m / (l_m * l_m)) * t_2) * k) * k);
	} else {
		tmp = 2.0 / ((t_2 * ((k / l_m) * (k / 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):
	t_2 = math.pow(math.sin(k), 2.0) / math.cos(k)
	tmp = 0
	if (l_m * l_m) <= 4e+123:
		tmp = 2.0 / ((((t_m / (l_m * l_m)) * t_2) * k) * k)
	else:
		tmp = 2.0 / ((t_2 * ((k / l_m) * (k / 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)
	t_2 = Float64((sin(k) ^ 2.0) / cos(k))
	tmp = 0.0
	if (Float64(l_m * l_m) <= 4e+123)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / Float64(l_m * l_m)) * t_2) * k) * k));
	else
		tmp = Float64(2.0 / Float64(Float64(t_2 * Float64(Float64(k / l_m) * Float64(k / 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)
	t_2 = (sin(k) ^ 2.0) / cos(k);
	tmp = 0.0;
	if ((l_m * l_m) <= 4e+123)
		tmp = 2.0 / ((((t_m / (l_m * l_m)) * t_2) * k) * k);
	else
		tmp = 2.0 / ((t_2 * ((k / l_m) * (k / 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_] := Block[{t$95$2 = N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 4e+123], N[(2.0 / N[(N[(N[(N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$2 * N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 3.99999999999999991e123

   1. Initial program 63.1%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 65.8

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

   5. Applied rewrites 65.8%

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

   7. Applied rewrites 75.5%

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

   if 3.99999999999999991e123 < (*.f64 l l)

   1. Initial program 41.3%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 62.8%

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

   6. Taylor expanded in t around 0

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

      1. frac-times N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-/.f64 54.9

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. times-frac N/A

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

   8. Applied rewrites 73.8%

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

Alternative 8: 70.5% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right)\right) \cdot t\_m} \end{array} \]

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
  (/ 2.0 (* (* (/ (pow (sin k) 2.0) (cos k)) (* (/ k l_m) (/ k 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) {
	return t_s * (2.0 / (((pow(sin(k), 2.0) / cos(k)) * ((k / l_m) * (k / l_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 * (2.0d0 / ((((sin(k) ** 2.0d0) / cos(k)) * ((k / l_m) * (k / l_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 * (2.0 / (((Math.pow(Math.sin(k), 2.0) / Math.cos(k)) * ((k / l_m) * (k / l_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 * (2.0 / (((math.pow(math.sin(k), 2.0) / math.cos(k)) * ((k / l_m) * (k / l_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(2.0 / Float64(Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(k / l_m) * Float64(k / l_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 * (2.0 / ((((sin(k) ^ 2.0) / cos(k)) * ((k / l_m) * (k / l_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[(2.0 / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.3%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 71.0%

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

6. Taylor expanded in t around 0

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

   1. frac-times N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-pow.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. lift-/.f64 62.0

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

   13. lift-/.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. times-frac N/A

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

8. Applied rewrites 71.4%

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

Alternative 9: 64.8% accurate, N/A× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\frac{\left(\left(l\_m \cdot l\_m\right) \cdot 2\right) \cdot \cos k}{k}}{\left({\sin k}^{2} \cdot t\_m\right) \cdot k} \end{array} \]

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) 2.0) (cos k)) k) (* (* (pow (sin k) 2.0) t_m) 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) {
	return t_s * (((((l_m * l_m) * 2.0) * cos(k)) / k) / ((pow(sin(k), 2.0) * t_m) * k));
}

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) * 2.0d0) * cos(k)) / k) / (((sin(k) ** 2.0d0) * t_m) * k))
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) * 2.0) * Math.cos(k)) / k) / ((Math.pow(Math.sin(k), 2.0) * t_m) * k));
}

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) * 2.0) * math.cos(k)) / k) / ((math.pow(math.sin(k), 2.0) * t_m) * k))

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(Float64(Float64(Float64(l_m * l_m) * 2.0) * cos(k)) / k) / Float64(Float64((sin(k) ^ 2.0) * t_m) * k)))
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) * 2.0) * cos(k)) / k) / (((sin(k) ^ 2.0) * t_m) * k));
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[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.3%

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

3. Taylor expanded in t around 0

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

   1. count-2-rev N/A

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

   2. div-add-rev N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*r* N/A

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

   15. lower-*.f64 N/A

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

5. Applied rewrites 60.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 63.4

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

7. Applied rewrites 63.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. lift-pow.f64 63.4

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

9. Applied rewrites 63.4%

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

    1. lift-/.f64 N/A

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

    2. lift-cos.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-fma.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. lift-cos.f64 N/A

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

    8. lift-*.f64 N/A

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

    9. lift-*.f64 N/A

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

    10. lift-*.f64 N/A

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

    11. lift-pow.f64 N/A

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

    12. lift-sin.f64 N/A

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

    13. associate-/r* N/A

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

    14. lower-/.f64 N/A

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

11. Applied rewrites 66.9%

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

Reproduce

herbie shell --seed 2025065 
(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))))