Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.4% → 92.4%

Time: 7.9s

Alternatives: 5

Speedup: N/A×

• 39.6% 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: 35.4% 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: 92.4% accurate, N/A× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.45 \cdot 10^{-129}:\\ \;\;\;\;\left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(\left(t\_2 \cdot -1\right) \cdot t\_m\right) \cdot k\right) \cdot k}\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos k}{t\_m} \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot 2\right)}{t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (sin k) 2.0)))
   (*
    t_s
    (if (<= t_m 1.45e-129)
      (* (* l (* l (/ (cos k) (* (* (* (* t_2 -1.0) t_m) k) k)))) -2.0)
      (/ (* (/ (cos k) t_m) (* (pow (/ l k) 2.0) 2.0)) t_2)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(sin(k), 2.0);
	double tmp;
	if (t_m <= 1.45e-129) {
		tmp = (l * (l * (cos(k) / ((((t_2 * -1.0) * t_m) * k) * k)))) * -2.0;
	} else {
		tmp = ((cos(k) / t_m) * (pow((l / k), 2.0) * 2.0)) / t_2;
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sin(k) ** 2.0d0
    if (t_m <= 1.45d-129) then
        tmp = (l * (l * (cos(k) / ((((t_2 * (-1.0d0)) * t_m) * k) * k)))) * (-2.0d0)
    else
        tmp = ((cos(k) / t_m) * (((l / k) ** 2.0d0) * 2.0d0)) / t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (t_m <= 1.45e-129) {
		tmp = (l * (l * (Math.cos(k) / ((((t_2 * -1.0) * t_m) * k) * k)))) * -2.0;
	} else {
		tmp = ((Math.cos(k) / t_m) * (Math.pow((l / k), 2.0) * 2.0)) / t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if t_m <= 1.45e-129:
		tmp = (l * (l * (math.cos(k) / ((((t_2 * -1.0) * t_m) * k) * k)))) * -2.0
	else:
		tmp = ((math.cos(k) / t_m) * (math.pow((l / k), 2.0) * 2.0)) / t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = sin(k) ^ 2.0
	tmp = 0.0
	if (t_m <= 1.45e-129)
		tmp = Float64(Float64(l * Float64(l * Float64(cos(k) / Float64(Float64(Float64(Float64(t_2 * -1.0) * t_m) * k) * k)))) * -2.0);
	else
		tmp = Float64(Float64(Float64(cos(k) / t_m) * Float64((Float64(l / k) ^ 2.0) * 2.0)) / t_2);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (t_m <= 1.45e-129)
		tmp = (l * (l * (cos(k) / ((((t_2 * -1.0) * t_m) * k) * k)))) * -2.0;
	else
		tmp = ((cos(k) / t_m) * (((l / k) ^ 2.0) * 2.0)) / t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.45e-129], N[(N[(l * N[(l * N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(N[(t$95$2 * -1.0), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.45000000000000008e-129

   1. Initial program 27.0%

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

      1. lift-*.f64 N/A

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

      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. sqr-pow N/A

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

      5. sqr-neg-rev N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower-neg.f64 8.2

          \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{-\ell} \cdot \frac{{t}^{1.5}}{\color{blue}{-\ell}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   4. Applied rewrites 8.2%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{1.5}}{-\ell} \cdot \frac{{t}^{1.5}}{-\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval 8.2

         \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{-\ell} \cdot \frac{{t}^{0.75} \cdot {t}^{\color{blue}{0.75}}}{-\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   6. Applied rewrites 8.2%

      \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{-\ell} \cdot \frac{\color{blue}{{t}^{0.75} \cdot {t}^{0.75}}}{-\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   7. Taylor expanded in t around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 71.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-cos.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-pow.f64 N/A

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

       9. lift-sin.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. associate-*l* N/A

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

       12. lower-*.f64 N/A

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

       13. lower-*.f64 N/A

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

   11. Applied rewrites 85.3%

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

   if 1.45000000000000008e-129 < t

   1. Initial program 46.7%

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

   3. Taylor expanded in 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. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 75.9

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

   5. Applied rewrites 75.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. frac-times N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. associate-*r/ N/A

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

   7. Applied rewrites 76.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

   9. Applied rewrites 91.8%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-cos.f64 N/A

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

       5. lift-pow.f64 N/A

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

       6. lift-sin.f64 N/A

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

       7. associate-*l/ N/A

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

       8. lower-/.f64 N/A

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

   11. Applied rewrites 92.6%

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

Alternative 2: 91.6% accurate, N/A× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.2 \cdot 10^{-129}:\\ \;\;\;\;\left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(\left(t\_2 \cdot -1\right) \cdot t\_m\right) \cdot k\right) \cdot k}\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos k}{t\_m}}{t\_2} \cdot \mathsf{fma}\left(\frac{\ell}{k}, \frac{\ell}{k}, \frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (sin k) 2.0)))
   (*
    t_s
    (if (<= t_m 2.2e-129)
      (* (* l (* l (/ (cos k) (* (* (* (* t_2 -1.0) t_m) k) k)))) -2.0)
      (* (/ (/ (cos k) t_m) t_2) (fma (/ l k) (/ l k) (* (/ l k) (/ l k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(sin(k), 2.0);
	double tmp;
	if (t_m <= 2.2e-129) {
		tmp = (l * (l * (cos(k) / ((((t_2 * -1.0) * t_m) * k) * k)))) * -2.0;
	} else {
		tmp = ((cos(k) / t_m) / t_2) * fma((l / k), (l / k), ((l / k) * (l / k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = sin(k) ^ 2.0
	tmp = 0.0
	if (t_m <= 2.2e-129)
		tmp = Float64(Float64(l * Float64(l * Float64(cos(k) / Float64(Float64(Float64(Float64(t_2 * -1.0) * t_m) * k) * k)))) * -2.0);
	else
		tmp = Float64(Float64(Float64(cos(k) / t_m) / t_2) * fma(Float64(l / k), Float64(l / k), Float64(Float64(l / k) * Float64(l / k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.2e-129], N[(N[(l * N[(l * N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(N[(t$95$2 * -1.0), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision] + N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.20000000000000003e-129

   1. Initial program 27.0%

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

      1. lift-*.f64 N/A

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

      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. sqr-pow N/A

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

      5. sqr-neg-rev N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower-neg.f64 8.2

          \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{-\ell} \cdot \frac{{t}^{1.5}}{\color{blue}{-\ell}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   4. Applied rewrites 8.2%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{1.5}}{-\ell} \cdot \frac{{t}^{1.5}}{-\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval 8.2

         \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{-\ell} \cdot \frac{{t}^{0.75} \cdot {t}^{\color{blue}{0.75}}}{-\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   6. Applied rewrites 8.2%

      \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{-\ell} \cdot \frac{\color{blue}{{t}^{0.75} \cdot {t}^{0.75}}}{-\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   7. Taylor expanded in t around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 71.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-cos.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-pow.f64 N/A

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

       9. lift-sin.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. associate-*l* N/A

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

       12. lower-*.f64 N/A

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

       13. lower-*.f64 N/A

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

   11. Applied rewrites 85.3%

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

   if 2.20000000000000003e-129 < t

   1. Initial program 46.7%

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

   3. Taylor expanded in 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. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 75.9

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

   5. Applied rewrites 75.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. frac-times N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. associate-*r/ N/A

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

   7. Applied rewrites 76.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

   9. Applied rewrites 91.8%

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

Alternative 3: 85.9% accurate, N/A× speedup

Math

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

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (*
   (* l (* l (/ (cos k) (* (* (* (* (pow (sin k) 2.0) -1.0) t_m) k) k))))
   -2.0)))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * (l * (cos(k) / ((((pow(sin(k), 2.0) * -1.0) * t_m) * k) * k)))) * -2.0);
}

Fortran

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

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l * (l * (cos(k) / (((((sin(k) ** 2.0d0) * (-1.0d0)) * t_m) * k) * k)))) * (-2.0d0))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * (l * (Math.cos(k) / ((((Math.pow(Math.sin(k), 2.0) * -1.0) * t_m) * k) * k)))) * -2.0);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l * (l * (math.cos(k) / ((((math.pow(math.sin(k), 2.0) * -1.0) * t_m) * k) * k)))) * -2.0)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l * Float64(l * Float64(cos(k) / Float64(Float64(Float64(Float64((sin(k) ^ 2.0) * -1.0) * t_m) * k) * k)))) * -2.0))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l * (l * (cos(k) / (((((sin(k) ^ 2.0) * -1.0) * t_m) * k) * k)))) * -2.0);
end

Wolfram

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

Derivation

1. Initial program 34.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. sqr-pow N/A

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

   5. sqr-neg-rev N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. metadata-eval N/A

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

   15. lower-neg.f64 25.4

       \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{-\ell} \cdot \frac{{t}^{1.5}}{\color{blue}{-\ell}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

4. Applied rewrites 25.4%

   \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{1.5}}{-\ell} \cdot \frac{{t}^{1.5}}{-\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. metadata-eval N/A

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

   6. lower-pow.f64 N/A

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

   7. metadata-eval 25.4

      \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{-\ell} \cdot \frac{{t}^{0.75} \cdot {t}^{\color{blue}{0.75}}}{-\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

6. Applied rewrites 25.4%

   \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{-\ell} \cdot \frac{\color{blue}{{t}^{0.75} \cdot {t}^{0.75}}}{-\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

7. Taylor expanded in t around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

9. Applied rewrites 72.9%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. lift-cos.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. lift-pow.f64 N/A

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

    9. lift-sin.f64 N/A

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

    10. lift-*.f64 N/A

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

    11. associate-*l* N/A

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

    12. lower-*.f64 N/A

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

    13. lower-*.f64 N/A

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

11. Applied rewrites 85.3%

    \[\leadsto \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(\left({\sin k}^{2} \cdot -1\right) \cdot t\right) \cdot k\right) \cdot k}\right)\right) \cdot -2 \]
12. Add Preprocessing

Alternative 4: 31.7% accurate, N/A× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot t\_m}{k \cdot k}\\ t_3 := \frac{\cos k}{\frac{\frac{{t\_m}^{3}}{\ell}}{\ell} \cdot {\sin k}^{2}}\\ t\_s \cdot \mathsf{fma}\left(t\_2, t\_3, t\_2 \cdot t\_3\right) \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (* t_m t_m) (* k k)))
        (t_3 (/ (cos k) (* (/ (/ (pow t_m 3.0) l) l) (pow (sin k) 2.0)))))
   (* t_s (fma t_2 t_3 (* t_2 t_3)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m * t_m) / (k * k);
	double t_3 = cos(k) / (((pow(t_m, 3.0) / l) / l) * pow(sin(k), 2.0));
	return t_s * fma(t_2, t_3, (t_2 * t_3));
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(t_m * t_m) / Float64(k * k))
	t_3 = Float64(cos(k) / Float64(Float64(Float64((t_m ^ 3.0) / l) / l) * (sin(k) ^ 2.0)))
	return Float64(t_s * fma(t_2, t_3, Float64(t_2 * t_3)))
end

Wolfram

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

Derivation

1. Initial program 34.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow-to-exp N/A

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

   5. pow2 N/A

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

   6. pow-to-exp N/A

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

   7. div-exp N/A

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

   8. lower-exp.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-log.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-log.f64 10.4

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

4. Applied rewrites 10.4%

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

5. Taylor expanded in t around 0

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

   1. count-2-rev N/A

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

   2. times-frac N/A

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

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{{t}^{2}}{{k}^{2}}, \color{blue}{\frac{\cos k}{e^{3 \cdot \log t - 2 \cdot \log \ell} \cdot {\sin k}^{2}}}, \frac{{t}^{2} \cdot \cos k}{{k}^{2} \cdot \left(e^{3 \cdot \log t - 2 \cdot \log \ell} \cdot {\sin k}^{2}\right)}\right) \]

7. Applied rewrites 33.8%

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

8. Final simplification 33.8%

   \[\leadsto \mathsf{fma}\left(\frac{t \cdot t}{k \cdot k}, \frac{\cos k}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot {\sin k}^{2}}, \frac{t \cdot t}{k \cdot k} \cdot \frac{\cos k}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot {\sin k}^{2}}\right) \]
9. Add Preprocessing

Alternative 5: 11.9% accurate, N/A× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\frac{{t\_m}^{3}}{\ell}}{\ell}\\ t\_s \cdot \frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \frac{t\_m \cdot t\_m}{t\_2}, -0.3333333333333333, \frac{2 \cdot \left(t\_m \cdot t\_m\right)}{t\_2}\right)}{{\left(k \cdot k\right)}^{2}} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (/ (pow t_m 3.0) l) l)))
   (*
    t_s
    (/
     (fma
      (* (* k k) (/ (* t_m t_m) t_2))
      -0.3333333333333333
      (/ (* 2.0 (* t_m t_m)) t_2))
     (pow (* k k) 2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (pow(t_m, 3.0) / l) / l;
	return t_s * (fma(((k * k) * ((t_m * t_m) / t_2)), -0.3333333333333333, ((2.0 * (t_m * t_m)) / t_2)) / pow((k * k), 2.0));
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64((t_m ^ 3.0) / l) / l)
	return Float64(t_s * Float64(fma(Float64(Float64(k * k) * Float64(Float64(t_m * t_m) / t_2)), -0.3333333333333333, Float64(Float64(2.0 * Float64(t_m * t_m)) / t_2)) / (Float64(k * k) ^ 2.0)))
end

Wolfram

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

Derivation

1. Initial program 34.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow-to-exp N/A

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

   5. pow2 N/A

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

   6. pow-to-exp N/A

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

   7. div-exp N/A

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

   8. lower-exp.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-log.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-log.f64 10.4

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

4. Applied rewrites 10.4%

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

5. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {t}^{2}}{e^{3 \cdot \log t - 2 \cdot \log \ell}} + 2 \cdot \frac{{t}^{2}}{e^{3 \cdot \log t - 2 \cdot \log \ell}}}{{k}^{4}}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {t}^{2}}{e^{3 \cdot \log t - 2 \cdot \log \ell}} + 2 \cdot \frac{{t}^{2}}{e^{3 \cdot \log t - 2 \cdot \log \ell}}}{\color{blue}{{k}^{4}}} \]

7. Applied rewrites 11.6%

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

8. Final simplification 11.6%

   \[\leadsto \frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \frac{t \cdot t}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}, -0.3333333333333333, \frac{2 \cdot \left(t \cdot t\right)}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)}{{\left(k \cdot k\right)}^{2}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025066 
(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))))