Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.4% → 90.2%

Time: 7.5s

Alternatives: 18

Speedup: 6.6×

• 72.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.4% accurate, 1.0× speedup

Math

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

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: 90.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|k\right|\right)\\ t_2 := \tan \left(\left|k\right|\right)\\ t_3 := \left|k\right| \cdot t\\ \mathbf{if}\;\left|k\right| \leq 4 \cdot 10^{-156}:\\ \;\;\;\;\frac{\ell}{\left(t\_3 \cdot t\right) \cdot t\_3} \cdot \ell\\ \mathbf{elif}\;\left|k\right| \leq 4 \cdot 10^{+114}:\\ \;\;\;\;\frac{2}{t \cdot \mathsf{fma}\left(\frac{\frac{t}{\ell} \cdot t}{\ell}, \left(2 \cdot t\_1\right) \cdot t\_2, \left(t\_2 \cdot \frac{t\_1}{\ell}\right) \cdot \frac{\left|k\right| \cdot \left|k\right|}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\cos \left(\left|k\right|\right) \cdot \ell}{\left|k\right|} \cdot \frac{\ell}{\left|k\right|}\right) \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (sin (fabs k))) (t_2 (tan (fabs k))) (t_3 (* (fabs k) t)))
   (if (<= (fabs k) 4e-156)
     (* (/ l (* (* t_3 t) t_3)) l)
     (if (<= (fabs k) 4e+114)
       (/
        2.0
        (*
         t
         (fma
          (/ (* (/ t l) t) l)
          (* (* 2.0 t_1) t_2)
          (* (* t_2 (/ t_1 l)) (/ (* (fabs k) (fabs k)) l)))))
       (*
        (* (/ (* (cos (fabs k)) l) (fabs k)) (/ l (fabs k)))
        (/ (/ 2.0 t) (fma (cos (+ (fabs k) (fabs k))) -0.5 0.5)))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(fabs(k));
	double t_2 = tan(fabs(k));
	double t_3 = fabs(k) * t;
	double tmp;
	if (fabs(k) <= 4e-156) {
		tmp = (l / ((t_3 * t) * t_3)) * l;
	} else if (fabs(k) <= 4e+114) {
		tmp = 2.0 / (t * fma((((t / l) * t) / l), ((2.0 * t_1) * t_2), ((t_2 * (t_1 / l)) * ((fabs(k) * fabs(k)) / l))));
	} else {
		tmp = (((cos(fabs(k)) * l) / fabs(k)) * (l / fabs(k))) * ((2.0 / t) / fma(cos((fabs(k) + fabs(k))), -0.5, 0.5));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = sin(abs(k))
	t_2 = tan(abs(k))
	t_3 = Float64(abs(k) * t)
	tmp = 0.0
	if (abs(k) <= 4e-156)
		tmp = Float64(Float64(l / Float64(Float64(t_3 * t) * t_3)) * l);
	elseif (abs(k) <= 4e+114)
		tmp = Float64(2.0 / Float64(t * fma(Float64(Float64(Float64(t / l) * t) / l), Float64(Float64(2.0 * t_1) * t_2), Float64(Float64(t_2 * Float64(t_1 / l)) * Float64(Float64(abs(k) * abs(k)) / l)))));
	else
		tmp = Float64(Float64(Float64(Float64(cos(abs(k)) * l) / abs(k)) * Float64(l / abs(k))) * Float64(Float64(2.0 / t) / fma(cos(Float64(abs(k) + abs(k))), -0.5, 0.5)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 4e-156], N[(N[(l / N[(N[(t$95$3 * t), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 4e+114], N[(2.0 / N[(t * N[(N[(N[(N[(t / l), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * N[(N[(2.0 * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(N[(t$95$2 * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] / N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / t), $MachinePrecision] / N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.0000000000000002e-156

   1. Initial program 54.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 50.8%

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

   4. Applied rewrites 50.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.2%

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow3 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 59.5%

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

   6. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.5%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 62.6%

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

   8. Applied rewrites 62.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 66.2%

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

   10. Applied rewrites 66.2%

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

   if 4.0000000000000002e-156 < k < 4.0000000000000001e114

   1. Initial program 54.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 62.8%

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

      1. lift-fma.f64 N/A

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

   6. Applied rewrites 68.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 71.5%

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

   8. Applied rewrites 71.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 77.3%

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

   10. Applied rewrites 77.3%

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

   if 4.0000000000000001e114 < k

   1. Initial program 54.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 60.5%

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

   4. Applied rewrites 60.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   6. Applied rewrites 57.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 60.4%

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

      14. lift--.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

   8. Applied rewrites 60.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 66.6%

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

   10. Applied rewrites 66.6%

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

Alternative 2: 88.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|k\right|\right)\\ t_2 := \tan \left(\left|k\right|\right)\\ t_3 := \left|k\right| \cdot t\\ \mathbf{if}\;\left|k\right| \leq 4.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{\ell}{\left(t\_3 \cdot t\right) \cdot t\_3} \cdot \ell\\ \mathbf{elif}\;\left|k\right| \leq 1.25 \cdot 10^{+99}:\\ \;\;\;\;\frac{2}{t \cdot \mathsf{fma}\left(\frac{\frac{t}{\ell} \cdot t}{\ell}, \left(2 \cdot t\_1\right) \cdot t\_2, \frac{t\_2 \cdot t\_1}{\ell \cdot \ell} \cdot \left(\left|k\right| \cdot \left|k\right|\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\cos \left(\left|k\right|\right) \cdot \ell}{\left|k\right|} \cdot \frac{\ell}{\left|k\right|}\right) \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (sin (fabs k))) (t_2 (tan (fabs k))) (t_3 (* (fabs k) t)))
   (if (<= (fabs k) 4.6e-109)
     (* (/ l (* (* t_3 t) t_3)) l)
     (if (<= (fabs k) 1.25e+99)
       (/
        2.0
        (*
         t
         (fma
          (/ (* (/ t l) t) l)
          (* (* 2.0 t_1) t_2)
          (* (/ (* t_2 t_1) (* l l)) (* (fabs k) (fabs k))))))
       (*
        (* (/ (* (cos (fabs k)) l) (fabs k)) (/ l (fabs k)))
        (/ (/ 2.0 t) (fma (cos (+ (fabs k) (fabs k))) -0.5 0.5)))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(fabs(k));
	double t_2 = tan(fabs(k));
	double t_3 = fabs(k) * t;
	double tmp;
	if (fabs(k) <= 4.6e-109) {
		tmp = (l / ((t_3 * t) * t_3)) * l;
	} else if (fabs(k) <= 1.25e+99) {
		tmp = 2.0 / (t * fma((((t / l) * t) / l), ((2.0 * t_1) * t_2), (((t_2 * t_1) / (l * l)) * (fabs(k) * fabs(k)))));
	} else {
		tmp = (((cos(fabs(k)) * l) / fabs(k)) * (l / fabs(k))) * ((2.0 / t) / fma(cos((fabs(k) + fabs(k))), -0.5, 0.5));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = sin(abs(k))
	t_2 = tan(abs(k))
	t_3 = Float64(abs(k) * t)
	tmp = 0.0
	if (abs(k) <= 4.6e-109)
		tmp = Float64(Float64(l / Float64(Float64(t_3 * t) * t_3)) * l);
	elseif (abs(k) <= 1.25e+99)
		tmp = Float64(2.0 / Float64(t * fma(Float64(Float64(Float64(t / l) * t) / l), Float64(Float64(2.0 * t_1) * t_2), Float64(Float64(Float64(t_2 * t_1) / Float64(l * l)) * Float64(abs(k) * abs(k))))));
	else
		tmp = Float64(Float64(Float64(Float64(cos(abs(k)) * l) / abs(k)) * Float64(l / abs(k))) * Float64(Float64(2.0 / t) / fma(cos(Float64(abs(k) + abs(k))), -0.5, 0.5)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 4.6e-109], N[(N[(l / N[(N[(t$95$3 * t), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 1.25e+99], N[(2.0 / N[(t * N[(N[(N[(N[(t / l), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * N[(N[(2.0 * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(N[(N[(t$95$2 * t$95$1), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] / N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / t), $MachinePrecision] / N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.6000000000000003e-109

   1. Initial program 54.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 50.8%

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

   4. Applied rewrites 50.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.2%

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow3 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 59.5%

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

   6. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.5%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 62.6%

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

   8. Applied rewrites 62.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 66.2%

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

   10. Applied rewrites 66.2%

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

   if 4.6000000000000003e-109 < k < 1.25e99

   1. Initial program 54.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 62.8%

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

      1. lift-fma.f64 N/A

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

   6. Applied rewrites 68.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 71.5%

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

   8. Applied rewrites 71.5%

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

   if 1.25e99 < k

   1. Initial program 54.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 60.5%

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

   4. Applied rewrites 60.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   6. Applied rewrites 57.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 60.4%

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

      14. lift--.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

   8. Applied rewrites 60.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 66.6%

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

   10. Applied rewrites 66.6%

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

Alternative 3: 85.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \cos \left(\left|k\right|\right)\\ t_2 := \mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right)\\ t_3 := \left|k\right| \cdot t\\ \mathbf{if}\;\left|k\right| \leq 1.15 \cdot 10^{-114}:\\ \;\;\;\;\frac{\ell}{\left(t\_3 \cdot t\right) \cdot t\_3} \cdot \ell\\ \mathbf{elif}\;\left|k\right| \leq 1.45 \cdot 10^{-9}:\\ \;\;\;\;\left(\frac{2}{t} \cdot \frac{\ell}{t}\right) \cdot \frac{\ell}{\left(t \cdot \mathsf{fma}\left(\left|k\right|, \frac{\left|k\right|}{t \cdot t}, 2\right)\right) \cdot \left(\tan \left(\left|k\right|\right) \cdot \sin \left(\left|k\right|\right)\right)}\\ \mathbf{elif}\;\left|k\right| \leq 4.7 \cdot 10^{+144}:\\ \;\;\;\;t\_1 \cdot \left(\ell \cdot \frac{\ell + \ell}{\left(\left|k\right| \cdot \left|k\right|\right) \cdot \left(t\_2 \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t\_1 \cdot \ell}{\left|k\right|} \cdot \frac{\ell}{\left|k\right|}\right) \cdot \frac{\frac{2}{t}}{t\_2}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cos (fabs k)))
        (t_2 (fma (cos (+ (fabs k) (fabs k))) -0.5 0.5))
        (t_3 (* (fabs k) t)))
   (if (<= (fabs k) 1.15e-114)
     (* (/ l (* (* t_3 t) t_3)) l)
     (if (<= (fabs k) 1.45e-9)
       (*
        (* (/ 2.0 t) (/ l t))
        (/
         l
         (*
          (* t (fma (fabs k) (/ (fabs k) (* t t)) 2.0))
          (* (tan (fabs k)) (sin (fabs k))))))
       (if (<= (fabs k) 4.7e+144)
         (* t_1 (* l (/ (+ l l) (* (* (fabs k) (fabs k)) (* t_2 t)))))
         (* (* (/ (* t_1 l) (fabs k)) (/ l (fabs k))) (/ (/ 2.0 t) t_2)))))))

C

double code(double t, double l, double k) {
	double t_1 = cos(fabs(k));
	double t_2 = fma(cos((fabs(k) + fabs(k))), -0.5, 0.5);
	double t_3 = fabs(k) * t;
	double tmp;
	if (fabs(k) <= 1.15e-114) {
		tmp = (l / ((t_3 * t) * t_3)) * l;
	} else if (fabs(k) <= 1.45e-9) {
		tmp = ((2.0 / t) * (l / t)) * (l / ((t * fma(fabs(k), (fabs(k) / (t * t)), 2.0)) * (tan(fabs(k)) * sin(fabs(k)))));
	} else if (fabs(k) <= 4.7e+144) {
		tmp = t_1 * (l * ((l + l) / ((fabs(k) * fabs(k)) * (t_2 * t))));
	} else {
		tmp = (((t_1 * l) / fabs(k)) * (l / fabs(k))) * ((2.0 / t) / t_2);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cos(abs(k))
	t_2 = fma(cos(Float64(abs(k) + abs(k))), -0.5, 0.5)
	t_3 = Float64(abs(k) * t)
	tmp = 0.0
	if (abs(k) <= 1.15e-114)
		tmp = Float64(Float64(l / Float64(Float64(t_3 * t) * t_3)) * l);
	elseif (abs(k) <= 1.45e-9)
		tmp = Float64(Float64(Float64(2.0 / t) * Float64(l / t)) * Float64(l / Float64(Float64(t * fma(abs(k), Float64(abs(k) / Float64(t * t)), 2.0)) * Float64(tan(abs(k)) * sin(abs(k))))));
	elseif (abs(k) <= 4.7e+144)
		tmp = Float64(t_1 * Float64(l * Float64(Float64(l + l) / Float64(Float64(abs(k) * abs(k)) * Float64(t_2 * t)))));
	else
		tmp = Float64(Float64(Float64(Float64(t_1 * l) / abs(k)) * Float64(l / abs(k))) * Float64(Float64(2.0 / t) / t_2));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 1.15e-114], N[(N[(l / N[(N[(t$95$3 * t), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 1.45e-9], N[(N[(N[(2.0 / t), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(t * N[(N[Abs[k], $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 4.7e+144], N[(t$95$1 * N[(l * N[(N[(l + l), $MachinePrecision] / N[(N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$1 * l), $MachinePrecision] / N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / t), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 1.15e-114

   1. Initial program 54.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 50.8%

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

   4. Applied rewrites 50.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.2%

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow3 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 59.5%

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

   6. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.5%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 62.6%

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

   8. Applied rewrites 62.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 66.2%

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

   10. Applied rewrites 66.2%

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

   if 1.15e-114 < k < 1.45e-9

   1. Initial program 54.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-/r* N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 50.6%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 50.6%

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 54.1%

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

   5. Applied rewrites 54.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

   7. Applied rewrites 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 60.6%

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

   9. Applied rewrites 60.6%

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

   if 1.45e-9 < k < 4.7000000000000002e144

   1. Initial program 54.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 60.5%

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

   4. Applied rewrites 60.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   6. Applied rewrites 57.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 60.4%

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

      14. lift--.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

   8. Applied rewrites 60.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/l/ N/A

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

      12. frac-times N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

   10. Applied rewrites 62.8%

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

   if 4.7000000000000002e144 < k

   1. Initial program 54.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 60.5%

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

   4. Applied rewrites 60.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   6. Applied rewrites 57.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 60.4%

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

      14. lift--.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

   8. Applied rewrites 60.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 66.6%

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

   10. Applied rewrites 66.6%

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

Alternative 4: 85.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \cos \left(\left|k\right|\right)\\ t_2 := \mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right)\\ t_3 := \left|k\right| \cdot t\\ \mathbf{if}\;\left|k\right| \leq 1.15 \cdot 10^{-114}:\\ \;\;\;\;\frac{\ell}{\left(t\_3 \cdot t\right) \cdot t\_3} \cdot \ell\\ \mathbf{elif}\;\left|k\right| \leq 1.45 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\ell + \ell}{t}}{t} \cdot \frac{\ell}{\left(t \cdot \mathsf{fma}\left(\left|k\right|, \frac{\left|k\right|}{t \cdot t}, 2\right)\right) \cdot \left(\tan \left(\left|k\right|\right) \cdot \sin \left(\left|k\right|\right)\right)}\\ \mathbf{elif}\;\left|k\right| \leq 4.7 \cdot 10^{+144}:\\ \;\;\;\;t\_1 \cdot \left(\ell \cdot \frac{\ell + \ell}{\left(\left|k\right| \cdot \left|k\right|\right) \cdot \left(t\_2 \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t\_1 \cdot \ell}{\left|k\right|} \cdot \frac{\ell}{\left|k\right|}\right) \cdot \frac{\frac{2}{t}}{t\_2}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cos (fabs k)))
        (t_2 (fma (cos (+ (fabs k) (fabs k))) -0.5 0.5))
        (t_3 (* (fabs k) t)))
   (if (<= (fabs k) 1.15e-114)
     (* (/ l (* (* t_3 t) t_3)) l)
     (if (<= (fabs k) 1.45e-9)
       (*
        (/ (/ (+ l l) t) t)
        (/
         l
         (*
          (* t (fma (fabs k) (/ (fabs k) (* t t)) 2.0))
          (* (tan (fabs k)) (sin (fabs k))))))
       (if (<= (fabs k) 4.7e+144)
         (* t_1 (* l (/ (+ l l) (* (* (fabs k) (fabs k)) (* t_2 t)))))
         (* (* (/ (* t_1 l) (fabs k)) (/ l (fabs k))) (/ (/ 2.0 t) t_2)))))))

C

double code(double t, double l, double k) {
	double t_1 = cos(fabs(k));
	double t_2 = fma(cos((fabs(k) + fabs(k))), -0.5, 0.5);
	double t_3 = fabs(k) * t;
	double tmp;
	if (fabs(k) <= 1.15e-114) {
		tmp = (l / ((t_3 * t) * t_3)) * l;
	} else if (fabs(k) <= 1.45e-9) {
		tmp = (((l + l) / t) / t) * (l / ((t * fma(fabs(k), (fabs(k) / (t * t)), 2.0)) * (tan(fabs(k)) * sin(fabs(k)))));
	} else if (fabs(k) <= 4.7e+144) {
		tmp = t_1 * (l * ((l + l) / ((fabs(k) * fabs(k)) * (t_2 * t))));
	} else {
		tmp = (((t_1 * l) / fabs(k)) * (l / fabs(k))) * ((2.0 / t) / t_2);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cos(abs(k))
	t_2 = fma(cos(Float64(abs(k) + abs(k))), -0.5, 0.5)
	t_3 = Float64(abs(k) * t)
	tmp = 0.0
	if (abs(k) <= 1.15e-114)
		tmp = Float64(Float64(l / Float64(Float64(t_3 * t) * t_3)) * l);
	elseif (abs(k) <= 1.45e-9)
		tmp = Float64(Float64(Float64(Float64(l + l) / t) / t) * Float64(l / Float64(Float64(t * fma(abs(k), Float64(abs(k) / Float64(t * t)), 2.0)) * Float64(tan(abs(k)) * sin(abs(k))))));
	elseif (abs(k) <= 4.7e+144)
		tmp = Float64(t_1 * Float64(l * Float64(Float64(l + l) / Float64(Float64(abs(k) * abs(k)) * Float64(t_2 * t)))));
	else
		tmp = Float64(Float64(Float64(Float64(t_1 * l) / abs(k)) * Float64(l / abs(k))) * Float64(Float64(2.0 / t) / t_2));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 1.15e-114], N[(N[(l / N[(N[(t$95$3 * t), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 1.45e-9], N[(N[(N[(N[(l + l), $MachinePrecision] / t), $MachinePrecision] / t), $MachinePrecision] * N[(l / N[(N[(t * N[(N[Abs[k], $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 4.7e+144], N[(t$95$1 * N[(l * N[(N[(l + l), $MachinePrecision] / N[(N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$1 * l), $MachinePrecision] / N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / t), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 1.15e-114

   1. Initial program 54.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 50.8%

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

   4. Applied rewrites 50.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.2%

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow3 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 59.5%

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

   6. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.5%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 62.6%

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

   8. Applied rewrites 62.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 66.2%

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

   10. Applied rewrites 66.2%

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

   if 1.15e-114 < k < 1.45e-9

   1. Initial program 54.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-/r* N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 50.6%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 50.6%

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 54.1%

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

   5. Applied rewrites 54.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

   7. Applied rewrites 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 60.6%

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

   9. Applied rewrites 60.6%

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

   if 1.45e-9 < k < 4.7000000000000002e144

   1. Initial program 54.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 60.5%

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

   4. Applied rewrites 60.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   6. Applied rewrites 57.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 60.4%

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

      14. lift--.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

   8. Applied rewrites 60.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/l/ N/A

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

      12. frac-times N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

   10. Applied rewrites 62.8%

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

   if 4.7000000000000002e144 < k

   1. Initial program 54.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 60.5%

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

   4. Applied rewrites 60.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   6. Applied rewrites 57.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 60.4%

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

      14. lift--.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

   8. Applied rewrites 60.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 66.6%

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

   10. Applied rewrites 66.6%

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

Alternative 5: 85.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \tan \left(\left|k\right|\right)\\ t_2 := \sin \left(\left|k\right|\right)\\ t_3 := \left|k\right| \cdot t\\ \mathbf{if}\;\left|k\right| \leq 5.8 \cdot 10^{-170}:\\ \;\;\;\;\frac{\ell}{\left(t\_3 \cdot t\right) \cdot t\_3} \cdot \ell\\ \mathbf{elif}\;\left|k\right| \leq 1.25 \cdot 10^{+99}:\\ \;\;\;\;\frac{2}{t \cdot \mathsf{fma}\left(t, \left(\frac{t}{\ell \cdot \ell} \cdot t\_1\right) \cdot \left(t\_2 \cdot 2\right), \left(\left|k\right| \cdot \left|k\right|\right) \cdot \frac{t\_1 \cdot t\_2}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\cos \left(\left|k\right|\right) \cdot \ell}{\left|k\right|} \cdot \frac{\ell}{\left|k\right|}\right) \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (tan (fabs k))) (t_2 (sin (fabs k))) (t_3 (* (fabs k) t)))
   (if (<= (fabs k) 5.8e-170)
     (* (/ l (* (* t_3 t) t_3)) l)
     (if (<= (fabs k) 1.25e+99)
       (/
        2.0
        (*
         t
         (fma
          t
          (* (* (/ t (* l l)) t_1) (* t_2 2.0))
          (* (* (fabs k) (fabs k)) (/ (* t_1 t_2) (* l l))))))
       (*
        (* (/ (* (cos (fabs k)) l) (fabs k)) (/ l (fabs k)))
        (/ (/ 2.0 t) (fma (cos (+ (fabs k) (fabs k))) -0.5 0.5)))))))

C

double code(double t, double l, double k) {
	double t_1 = tan(fabs(k));
	double t_2 = sin(fabs(k));
	double t_3 = fabs(k) * t;
	double tmp;
	if (fabs(k) <= 5.8e-170) {
		tmp = (l / ((t_3 * t) * t_3)) * l;
	} else if (fabs(k) <= 1.25e+99) {
		tmp = 2.0 / (t * fma(t, (((t / (l * l)) * t_1) * (t_2 * 2.0)), ((fabs(k) * fabs(k)) * ((t_1 * t_2) / (l * l)))));
	} else {
		tmp = (((cos(fabs(k)) * l) / fabs(k)) * (l / fabs(k))) * ((2.0 / t) / fma(cos((fabs(k) + fabs(k))), -0.5, 0.5));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = tan(abs(k))
	t_2 = sin(abs(k))
	t_3 = Float64(abs(k) * t)
	tmp = 0.0
	if (abs(k) <= 5.8e-170)
		tmp = Float64(Float64(l / Float64(Float64(t_3 * t) * t_3)) * l);
	elseif (abs(k) <= 1.25e+99)
		tmp = Float64(2.0 / Float64(t * fma(t, Float64(Float64(Float64(t / Float64(l * l)) * t_1) * Float64(t_2 * 2.0)), Float64(Float64(abs(k) * abs(k)) * Float64(Float64(t_1 * t_2) / Float64(l * l))))));
	else
		tmp = Float64(Float64(Float64(Float64(cos(abs(k)) * l) / abs(k)) * Float64(l / abs(k))) * Float64(Float64(2.0 / t) / fma(cos(Float64(abs(k) + abs(k))), -0.5, 0.5)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 5.8e-170], N[(N[(l / N[(N[(t$95$3 * t), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 1.25e+99], N[(2.0 / N[(t * N[(t * N[(N[(N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * t$95$2), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] / N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / t), $MachinePrecision] / N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.8000000000000001e-170

   1. Initial program 54.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 50.8%

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

   4. Applied rewrites 50.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.2%

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow3 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 59.5%

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

   6. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.5%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 62.6%

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

   8. Applied rewrites 62.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 66.2%

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

   10. Applied rewrites 66.2%

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

   if 5.8000000000000001e-170 < k < 1.25e99

   1. Initial program 54.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 62.8%

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

      1. lift-fma.f64 N/A

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

   6. Applied rewrites 68.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 71.0%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 71.0%

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

   8. Applied rewrites 71.0%

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

   if 1.25e99 < k

   1. Initial program 54.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 60.5%

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

   4. Applied rewrites 60.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   6. Applied rewrites 57.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 60.4%

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

      14. lift--.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

   8. Applied rewrites 60.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 66.6%

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

   10. Applied rewrites 66.6%

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

Alternative 6: 84.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \cos \left(\left|k\right|\right)\\ t_2 := \mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right)\\ t_3 := \left|k\right| \cdot t\\ \mathbf{if}\;\left|k\right| \leq 24:\\ \;\;\;\;\frac{\ell}{\left(t\_3 \cdot t\right) \cdot t\_3} \cdot \ell\\ \mathbf{elif}\;\left|k\right| \leq 4.7 \cdot 10^{+144}:\\ \;\;\;\;t\_1 \cdot \left(\ell \cdot \frac{\ell + \ell}{\left(\left|k\right| \cdot \left|k\right|\right) \cdot \left(t\_2 \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t\_1 \cdot \ell}{\left|k\right|} \cdot \frac{\ell}{\left|k\right|}\right) \cdot \frac{\frac{2}{t}}{t\_2}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cos (fabs k)))
        (t_2 (fma (cos (+ (fabs k) (fabs k))) -0.5 0.5))
        (t_3 (* (fabs k) t)))
   (if (<= (fabs k) 24.0)
     (* (/ l (* (* t_3 t) t_3)) l)
     (if (<= (fabs k) 4.7e+144)
       (* t_1 (* l (/ (+ l l) (* (* (fabs k) (fabs k)) (* t_2 t)))))
       (* (* (/ (* t_1 l) (fabs k)) (/ l (fabs k))) (/ (/ 2.0 t) t_2))))))

C

double code(double t, double l, double k) {
	double t_1 = cos(fabs(k));
	double t_2 = fma(cos((fabs(k) + fabs(k))), -0.5, 0.5);
	double t_3 = fabs(k) * t;
	double tmp;
	if (fabs(k) <= 24.0) {
		tmp = (l / ((t_3 * t) * t_3)) * l;
	} else if (fabs(k) <= 4.7e+144) {
		tmp = t_1 * (l * ((l + l) / ((fabs(k) * fabs(k)) * (t_2 * t))));
	} else {
		tmp = (((t_1 * l) / fabs(k)) * (l / fabs(k))) * ((2.0 / t) / t_2);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cos(abs(k))
	t_2 = fma(cos(Float64(abs(k) + abs(k))), -0.5, 0.5)
	t_3 = Float64(abs(k) * t)
	tmp = 0.0
	if (abs(k) <= 24.0)
		tmp = Float64(Float64(l / Float64(Float64(t_3 * t) * t_3)) * l);
	elseif (abs(k) <= 4.7e+144)
		tmp = Float64(t_1 * Float64(l * Float64(Float64(l + l) / Float64(Float64(abs(k) * abs(k)) * Float64(t_2 * t)))));
	else
		tmp = Float64(Float64(Float64(Float64(t_1 * l) / abs(k)) * Float64(l / abs(k))) * Float64(Float64(2.0 / t) / t_2));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 24.0], N[(N[(l / N[(N[(t$95$3 * t), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 4.7e+144], N[(t$95$1 * N[(l * N[(N[(l + l), $MachinePrecision] / N[(N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$1 * l), $MachinePrecision] / N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / t), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 24

   1. Initial program 54.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 50.8%

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

   4. Applied rewrites 50.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.2%

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow3 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 59.5%

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

   6. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.5%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 62.6%

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

   8. Applied rewrites 62.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 66.2%

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

   10. Applied rewrites 66.2%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   if 24 < k < 4.7000000000000002e144

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 60.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 60.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot t}}{\color{blue}{{\sin k}^{2}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot t}}{\color{blue}{{\sin k}^{2}}} \]

   6. Applied rewrites 57.5%

      \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{0.5 - 0.5 \cdot \cos \left(k + k\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      5. times-frac N/A

         \[\leadsto \frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \frac{2}{t}}{\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\frac{2}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\frac{2}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \frac{\frac{\color{blue}{2}}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      9. associate-/l* N/A

         \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\color{blue}{\frac{2}{t}}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\color{blue}{\frac{2}{t}}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{\color{blue}{t}}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      13. lower-/.f64 60.4%

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\color{blue}{0.5} - 0.5 \cdot \cos \left(k + k\right)} \]

      14. lift--.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(k + k\right)}} \]

      16. fp-cancel-sub-sign-inv N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(k + k\right)}} \]

   8. Applied rewrites 60.4%

      \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \color{blue}{\frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \color{blue}{\frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\color{blue}{\frac{2}{t}}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)} \]

      3. associate-*l* N/A

         \[\leadsto \left(\cos k \cdot \ell\right) \cdot \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\color{blue}{\frac{\ell}{k \cdot k}} \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}\right) \]

      5. associate-*l* N/A

         \[\leadsto \cos k \cdot \color{blue}{\left(\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \cos k \cdot \color{blue}{\left(\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \cos k \cdot \left(\ell \cdot \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \cos k \cdot \left(\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\color{blue}{\frac{2}{t}}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \cos k \cdot \left(\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\frac{2}{t}}{\color{blue}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \cos k \cdot \left(\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\color{blue}{\cos \left(k + k\right)}, \frac{-1}{2}, \frac{1}{2}\right)}\right)\right) \]

      11. associate-/l/ N/A

          \[\leadsto \cos k \cdot \left(\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{2}{\color{blue}{t \cdot \mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}}\right)\right) \]

      12. frac-times N/A

          \[\leadsto \cos k \cdot \left(\ell \cdot \frac{\ell \cdot 2}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      13. *-commutative N/A

          \[\leadsto \cos k \cdot \left(\ell \cdot \frac{2 \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \cos k \cdot \left(\ell \cdot \frac{2 \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   10. Applied rewrites 62.8%

       \[\leadsto \cos k \cdot \color{blue}{\left(\ell \cdot \frac{\ell + \ell}{\left(k \cdot k\right) \cdot \left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right)}\right)} \]

   if 4.7000000000000002e144 < k

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 60.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 60.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot t}}{\color{blue}{{\sin k}^{2}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot t}}{\color{blue}{{\sin k}^{2}}} \]

   6. Applied rewrites 57.5%

      \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{0.5 - 0.5 \cdot \cos \left(k + k\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      5. times-frac N/A

         \[\leadsto \frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \frac{2}{t}}{\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\frac{2}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\frac{2}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \frac{\frac{\color{blue}{2}}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      9. associate-/l* N/A

         \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\color{blue}{\frac{2}{t}}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\color{blue}{\frac{2}{t}}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{\color{blue}{t}}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      13. lower-/.f64 60.4%

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\color{blue}{0.5} - 0.5 \cdot \cos \left(k + k\right)} \]

      14. lift--.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(k + k\right)}} \]

      16. fp-cancel-sub-sign-inv N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(k + k\right)}} \]

   8. Applied rewrites 60.4%

      \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \color{blue}{\frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\color{blue}{\frac{2}{t}}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{\color{blue}{t}}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \frac{\color{blue}{\frac{2}{t}}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \frac{\frac{2}{\color{blue}{t}}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)} \]

      5. times-frac N/A

         \[\leadsto \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\color{blue}{\frac{2}{t}}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\color{blue}{\frac{2}{t}}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{\color{blue}{2}}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)} \]

      8. lower-/.f64 66.6%

         \[\leadsto \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{2}{\color{blue}{t}}}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)} \]

   10. Applied rewrites 66.6%

       \[\leadsto \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\color{blue}{\frac{2}{t}}}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 83.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \cos \left(\left|k\right|\right)\\ t_2 := \mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right)\\ t_3 := \left|k\right| \cdot t\\ \mathbf{if}\;\left|k\right| \leq 24:\\ \;\;\;\;\frac{\ell}{\left(t\_3 \cdot t\right) \cdot t\_3} \cdot \ell\\ \mathbf{elif}\;\left|k\right| \leq 4.7 \cdot 10^{+144}:\\ \;\;\;\;t\_1 \cdot \left(\ell \cdot \frac{\ell + \ell}{\left(\left|k\right| \cdot \left|k\right|\right) \cdot \left(t\_2 \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_1 \cdot \ell\right) \cdot \frac{\frac{\ell}{\left|k\right|}}{\left|k\right|}\right) \cdot \frac{\frac{2}{t}}{t\_2}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cos (fabs k)))
        (t_2 (fma (cos (+ (fabs k) (fabs k))) -0.5 0.5))
        (t_3 (* (fabs k) t)))
   (if (<= (fabs k) 24.0)
     (* (/ l (* (* t_3 t) t_3)) l)
     (if (<= (fabs k) 4.7e+144)
       (* t_1 (* l (/ (+ l l) (* (* (fabs k) (fabs k)) (* t_2 t)))))
       (* (* (* t_1 l) (/ (/ l (fabs k)) (fabs k))) (/ (/ 2.0 t) t_2))))))

C

double code(double t, double l, double k) {
	double t_1 = cos(fabs(k));
	double t_2 = fma(cos((fabs(k) + fabs(k))), -0.5, 0.5);
	double t_3 = fabs(k) * t;
	double tmp;
	if (fabs(k) <= 24.0) {
		tmp = (l / ((t_3 * t) * t_3)) * l;
	} else if (fabs(k) <= 4.7e+144) {
		tmp = t_1 * (l * ((l + l) / ((fabs(k) * fabs(k)) * (t_2 * t))));
	} else {
		tmp = ((t_1 * l) * ((l / fabs(k)) / fabs(k))) * ((2.0 / t) / t_2);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cos(abs(k))
	t_2 = fma(cos(Float64(abs(k) + abs(k))), -0.5, 0.5)
	t_3 = Float64(abs(k) * t)
	tmp = 0.0
	if (abs(k) <= 24.0)
		tmp = Float64(Float64(l / Float64(Float64(t_3 * t) * t_3)) * l);
	elseif (abs(k) <= 4.7e+144)
		tmp = Float64(t_1 * Float64(l * Float64(Float64(l + l) / Float64(Float64(abs(k) * abs(k)) * Float64(t_2 * t)))));
	else
		tmp = Float64(Float64(Float64(t_1 * l) * Float64(Float64(l / abs(k)) / abs(k))) * Float64(Float64(2.0 / t) / t_2));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 24.0], N[(N[(l / N[(N[(t$95$3 * t), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 4.7e+144], N[(t$95$1 * N[(l * N[(N[(l + l), $MachinePrecision] / N[(N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 * l), $MachinePrecision] * N[(N[(l / N[Abs[k], $MachinePrecision]), $MachinePrecision] / N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / t), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 24

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.8%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.2%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. pow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      12. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{{k}^{2}}} \]

      13. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

      14. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      15. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      17. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.5%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 59.5%

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      13. lower-*.f64 62.6%

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lower-*.f64 66.2%

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 66.2%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   if 24 < k < 4.7000000000000002e144

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 60.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 60.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot t}}{\color{blue}{{\sin k}^{2}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot t}}{\color{blue}{{\sin k}^{2}}} \]

   6. Applied rewrites 57.5%

      \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{0.5 - 0.5 \cdot \cos \left(k + k\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      5. times-frac N/A

         \[\leadsto \frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \frac{2}{t}}{\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\frac{2}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\frac{2}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \frac{\frac{\color{blue}{2}}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      9. associate-/l* N/A

         \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\color{blue}{\frac{2}{t}}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\color{blue}{\frac{2}{t}}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{\color{blue}{t}}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      13. lower-/.f64 60.4%

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\color{blue}{0.5} - 0.5 \cdot \cos \left(k + k\right)} \]

      14. lift--.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(k + k\right)}} \]

      16. fp-cancel-sub-sign-inv N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(k + k\right)}} \]

   8. Applied rewrites 60.4%

      \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \color{blue}{\frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \color{blue}{\frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\color{blue}{\frac{2}{t}}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)} \]

      3. associate-*l* N/A

         \[\leadsto \left(\cos k \cdot \ell\right) \cdot \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\color{blue}{\frac{\ell}{k \cdot k}} \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}\right) \]

      5. associate-*l* N/A

         \[\leadsto \cos k \cdot \color{blue}{\left(\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \cos k \cdot \color{blue}{\left(\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \cos k \cdot \left(\ell \cdot \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \cos k \cdot \left(\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\color{blue}{\frac{2}{t}}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \cos k \cdot \left(\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\frac{2}{t}}{\color{blue}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \cos k \cdot \left(\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\color{blue}{\cos \left(k + k\right)}, \frac{-1}{2}, \frac{1}{2}\right)}\right)\right) \]

      11. associate-/l/ N/A

          \[\leadsto \cos k \cdot \left(\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{2}{\color{blue}{t \cdot \mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}}\right)\right) \]

      12. frac-times N/A

          \[\leadsto \cos k \cdot \left(\ell \cdot \frac{\ell \cdot 2}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      13. *-commutative N/A

          \[\leadsto \cos k \cdot \left(\ell \cdot \frac{2 \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \cos k \cdot \left(\ell \cdot \frac{2 \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   10. Applied rewrites 62.8%

       \[\leadsto \cos k \cdot \color{blue}{\left(\ell \cdot \frac{\ell + \ell}{\left(k \cdot k\right) \cdot \left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right)}\right)} \]

   if 4.7000000000000002e144 < k

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 60.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 60.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot t}}{\color{blue}{{\sin k}^{2}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot t}}{\color{blue}{{\sin k}^{2}}} \]

   6. Applied rewrites 57.5%

      \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{0.5 - 0.5 \cdot \cos \left(k + k\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      5. times-frac N/A

         \[\leadsto \frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \frac{2}{t}}{\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\frac{2}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\frac{2}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \frac{\frac{\color{blue}{2}}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      9. associate-/l* N/A

         \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\color{blue}{\frac{2}{t}}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\color{blue}{\frac{2}{t}}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{\color{blue}{t}}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      13. lower-/.f64 60.4%

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\color{blue}{0.5} - 0.5 \cdot \cos \left(k + k\right)} \]

      14. lift--.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(k + k\right)}} \]

      16. fp-cancel-sub-sign-inv N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(k + k\right)}} \]

   8. Applied rewrites 60.4%

      \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \color{blue}{\frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{\color{blue}{t}}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)} \]

      3. associate-/r* N/A

         \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\frac{\ell}{k}}{k}\right) \cdot \frac{\frac{2}{\color{blue}{t}}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\frac{\ell}{k}}{k}\right) \cdot \frac{\frac{2}{\color{blue}{t}}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)} \]

      5. lower-/.f64 65.0%

         \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\frac{\ell}{k}}{k}\right) \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)} \]

   10. Applied rewrites 65.0%

       \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\frac{\ell}{k}}{k}\right) \cdot \frac{\frac{2}{\color{blue}{t}}}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 78.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \left|k\right| \cdot t\\ \mathbf{if}\;\left|k\right| \leq 24:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot t\right) \cdot t\_1} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left|k\right|\right) \cdot \left(\ell \cdot \frac{\ell + \ell}{\left(\left|k\right| \cdot \left|k\right|\right) \cdot \left(\mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right) \cdot t\right)}\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (fabs k) t)))
   (if (<= (fabs k) 24.0)
     (* (/ l (* (* t_1 t) t_1)) l)
     (*
      (cos (fabs k))
      (*
       l
       (/
        (+ l l)
        (*
         (* (fabs k) (fabs k))
         (* (fma (cos (+ (fabs k) (fabs k))) -0.5 0.5) t))))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(k) * t;
	double tmp;
	if (fabs(k) <= 24.0) {
		tmp = (l / ((t_1 * t) * t_1)) * l;
	} else {
		tmp = cos(fabs(k)) * (l * ((l + l) / ((fabs(k) * fabs(k)) * (fma(cos((fabs(k) + fabs(k))), -0.5, 0.5) * t))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(abs(k) * t)
	tmp = 0.0
	if (abs(k) <= 24.0)
		tmp = Float64(Float64(l / Float64(Float64(t_1 * t) * t_1)) * l);
	else
		tmp = Float64(cos(abs(k)) * Float64(l * Float64(Float64(l + l) / Float64(Float64(abs(k) * abs(k)) * Float64(fma(cos(Float64(abs(k) + abs(k))), -0.5, 0.5) * t)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 24.0], N[(N[(l / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] * N[(l * N[(N[(l + l), $MachinePrecision] / N[(N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 24

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.8%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.2%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. pow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      12. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{{k}^{2}}} \]

      13. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

      14. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      15. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      17. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.5%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 59.5%

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      13. lower-*.f64 62.6%

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lower-*.f64 66.2%

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 66.2%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   if 24 < k

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 60.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 60.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot t}}{\color{blue}{{\sin k}^{2}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot t}}{\color{blue}{{\sin k}^{2}}} \]

   6. Applied rewrites 57.5%

      \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{0.5 - 0.5 \cdot \cos \left(k + k\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      5. times-frac N/A

         \[\leadsto \frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \frac{2}{t}}{\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\frac{2}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\frac{2}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \frac{\frac{\color{blue}{2}}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      9. associate-/l* N/A

         \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\color{blue}{\frac{2}{t}}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\color{blue}{\frac{2}{t}}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{\color{blue}{t}}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      13. lower-/.f64 60.4%

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\color{blue}{0.5} - 0.5 \cdot \cos \left(k + k\right)} \]

      14. lift--.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(k + k\right)}} \]

      16. fp-cancel-sub-sign-inv N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(k + k\right)}} \]

   8. Applied rewrites 60.4%

      \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \color{blue}{\frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \color{blue}{\frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\color{blue}{\frac{2}{t}}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)} \]

      3. associate-*l* N/A

         \[\leadsto \left(\cos k \cdot \ell\right) \cdot \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\color{blue}{\frac{\ell}{k \cdot k}} \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}\right) \]

      5. associate-*l* N/A

         \[\leadsto \cos k \cdot \color{blue}{\left(\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \cos k \cdot \color{blue}{\left(\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \cos k \cdot \left(\ell \cdot \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \cos k \cdot \left(\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\color{blue}{\frac{2}{t}}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \cos k \cdot \left(\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\frac{2}{t}}{\color{blue}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \cos k \cdot \left(\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\color{blue}{\cos \left(k + k\right)}, \frac{-1}{2}, \frac{1}{2}\right)}\right)\right) \]

      11. associate-/l/ N/A

          \[\leadsto \cos k \cdot \left(\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{2}{\color{blue}{t \cdot \mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}}\right)\right) \]

      12. frac-times N/A

          \[\leadsto \cos k \cdot \left(\ell \cdot \frac{\ell \cdot 2}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      13. *-commutative N/A

          \[\leadsto \cos k \cdot \left(\ell \cdot \frac{2 \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \cos k \cdot \left(\ell \cdot \frac{2 \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}}\right) \]

   10. Applied rewrites 62.8%

       \[\leadsto \cos k \cdot \color{blue}{\left(\ell \cdot \frac{\ell + \ell}{\left(k \cdot k\right) \cdot \left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 78.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \left|k\right| \cdot t\\ \mathbf{if}\;\left|k\right| \leq 24:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot t\right) \cdot t\_1} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\cos \left(\left|k\right|\right) \cdot \frac{\ell + \ell}{\left(\left|k\right| \cdot \left|k\right|\right) \cdot \left(\mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right) \cdot t\right)}\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (fabs k) t)))
   (if (<= (fabs k) 24.0)
     (* (/ l (* (* t_1 t) t_1)) l)
     (*
      l
      (*
       (cos (fabs k))
       (/
        (+ l l)
        (*
         (* (fabs k) (fabs k))
         (* (fma (cos (+ (fabs k) (fabs k))) -0.5 0.5) t))))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(k) * t;
	double tmp;
	if (fabs(k) <= 24.0) {
		tmp = (l / ((t_1 * t) * t_1)) * l;
	} else {
		tmp = l * (cos(fabs(k)) * ((l + l) / ((fabs(k) * fabs(k)) * (fma(cos((fabs(k) + fabs(k))), -0.5, 0.5) * t))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(abs(k) * t)
	tmp = 0.0
	if (abs(k) <= 24.0)
		tmp = Float64(Float64(l / Float64(Float64(t_1 * t) * t_1)) * l);
	else
		tmp = Float64(l * Float64(cos(abs(k)) * Float64(Float64(l + l) / Float64(Float64(abs(k) * abs(k)) * Float64(fma(cos(Float64(abs(k) + abs(k))), -0.5, 0.5) * t)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 24.0], N[(N[(l / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(l * N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] * N[(N[(l + l), $MachinePrecision] / N[(N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 24

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.8%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.2%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. pow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      12. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{{k}^{2}}} \]

      13. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

      14. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      15. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      17. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.5%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 59.5%

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      13. lower-*.f64 62.6%

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lower-*.f64 66.2%

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 66.2%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   if 24 < k

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 60.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 60.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot t}}{\color{blue}{{\sin k}^{2}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot t}}{\color{blue}{{\sin k}^{2}}} \]

   6. Applied rewrites 57.5%

      \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{0.5 - 0.5 \cdot \cos \left(k + k\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      5. times-frac N/A

         \[\leadsto \frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \frac{2}{t}}{\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\frac{2}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\frac{2}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \frac{\frac{\color{blue}{2}}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      9. associate-/l* N/A

         \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\color{blue}{\frac{2}{t}}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\color{blue}{\frac{2}{t}}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{\color{blue}{t}}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      13. lower-/.f64 60.4%

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\color{blue}{0.5} - 0.5 \cdot \cos \left(k + k\right)} \]

      14. lift--.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(k + k\right)}} \]

      16. fp-cancel-sub-sign-inv N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(k + k\right)}} \]

   8. Applied rewrites 60.4%

      \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \color{blue}{\frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \color{blue}{\frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\color{blue}{\frac{2}{t}}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)} \]

      3. associate-*l* N/A

         \[\leadsto \left(\cos k \cdot \ell\right) \cdot \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\cos k \cdot \ell\right) \cdot \left(\color{blue}{\frac{\ell}{k \cdot k}} \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\ell \cdot \cos k\right) \cdot \left(\color{blue}{\frac{\ell}{k \cdot k}} \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}\right) \]

      6. associate-*l* N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\cos k \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\cos k \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \ell \cdot \left(\cos k \cdot \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}\right)}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \ell \cdot \left(\cos k \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\color{blue}{\frac{2}{t}}}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \ell \cdot \left(\cos k \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\frac{2}{t}}{\color{blue}{\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}}\right)\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \ell \cdot \left(\cos k \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\color{blue}{\cos \left(k + k\right)}, \frac{-1}{2}, \frac{1}{2}\right)}\right)\right) \]

      12. associate-/l/ N/A

          \[\leadsto \ell \cdot \left(\cos k \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{2}{\color{blue}{t \cdot \mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)}}\right)\right) \]

      13. frac-times N/A

          \[\leadsto \ell \cdot \left(\cos k \cdot \frac{\ell \cdot 2}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}}\right) \]

      14. *-commutative N/A

          \[\leadsto \ell \cdot \left(\cos k \cdot \frac{2 \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}\right) \]

   10. Applied rewrites 62.8%

       \[\leadsto \ell \cdot \color{blue}{\left(\cos k \cdot \frac{\ell + \ell}{\left(k \cdot k\right) \cdot \left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 75.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := k \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 4.1 \cdot 10^{+17}:\\ \;\;\;\;\frac{1}{\left(\left(k \cdot k\right) \cdot \left|t\right|\right) \cdot \frac{\tan k \cdot \sin k}{\ell \cdot \ell}} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot \left|t\right|\right) \cdot t\_1} \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 4.1e+17)
      (* (/ 1.0 (* (* (* k k) (fabs t)) (/ (* (tan k) (sin k)) (* l l)))) 2.0)
      (* (/ l (* (* t_1 (fabs t)) t_1)) l)))))

C

double code(double t, double l, double k) {
	double t_1 = k * fabs(t);
	double tmp;
	if (fabs(t) <= 4.1e+17) {
		tmp = (1.0 / (((k * k) * fabs(t)) * ((tan(k) * sin(k)) / (l * l)))) * 2.0;
	} else {
		tmp = (l / ((t_1 * fabs(t)) * t_1)) * l;
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = k * Math.abs(t);
	double tmp;
	if (Math.abs(t) <= 4.1e+17) {
		tmp = (1.0 / (((k * k) * Math.abs(t)) * ((Math.tan(k) * Math.sin(k)) / (l * l)))) * 2.0;
	} else {
		tmp = (l / ((t_1 * Math.abs(t)) * t_1)) * l;
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = k * math.fabs(t)
	tmp = 0
	if math.fabs(t) <= 4.1e+17:
		tmp = (1.0 / (((k * k) * math.fabs(t)) * ((math.tan(k) * math.sin(k)) / (l * l)))) * 2.0
	else:
		tmp = (l / ((t_1 * math.fabs(t)) * t_1)) * l
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * abs(t))
	tmp = 0.0
	if (abs(t) <= 4.1e+17)
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(k * k) * abs(t)) * Float64(Float64(tan(k) * sin(k)) / Float64(l * l)))) * 2.0);
	else
		tmp = Float64(Float64(l / Float64(Float64(t_1 * abs(t)) * t_1)) * l);
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * abs(t);
	tmp = 0.0;
	if (abs(t) <= 4.1e+17)
		tmp = (1.0 / (((k * k) * abs(t)) * ((tan(k) * sin(k)) / (l * l)))) * 2.0;
	else
		tmp = (l / ((t_1 * abs(t)) * t_1)) * l;
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 4.1e+17], N[(N[(1.0 / N[(N[(N[(k * k), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(l / N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.1e17

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 60.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 60.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Applied rewrites 61.3%

      \[\leadsto \color{blue}{\frac{1}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{\tan k \cdot \sin k}{\ell \cdot \ell}} \cdot 2} \]

   if 4.1e17 < t

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.8%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.2%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. pow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      12. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{{k}^{2}}} \]

      13. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

      14. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      15. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      17. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.5%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 59.5%

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      13. lower-*.f64 62.6%

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lower-*.f64 66.2%

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 66.2%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 72.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_1 := k \cdot \left|t\right|\\ t_2 := \left|t\right| \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 3.8 \cdot 10^{-160}:\\ \;\;\;\;\left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{2}{{k}^{2} \cdot \left|t\right|}\\ \mathbf{elif}\;\left|t\right| \leq 6 \cdot 10^{-25}:\\ \;\;\;\;\left(\frac{2}{t\_2} \cdot \ell\right) \cdot \frac{\ell}{\left(\left|t\right| \cdot \mathsf{fma}\left(k, \frac{k}{t\_2}, 2\right)\right) \cdot {k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot \left|t\right|\right) \cdot t\_1} \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (fabs t))) (t_2 (* (fabs t) (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 3.8e-160)
      (* (* (* (cos k) l) (/ l (* k k))) (/ 2.0 (* (pow k 2.0) (fabs t))))
      (if (<= (fabs t) 6e-25)
        (*
         (* (/ 2.0 t_2) l)
         (/ l (* (* (fabs t) (fma k (/ k t_2) 2.0)) (pow k 2.0))))
        (* (/ l (* (* t_1 (fabs t)) t_1)) l))))))

C

double code(double t, double l, double k) {
	double t_1 = k * fabs(t);
	double t_2 = fabs(t) * fabs(t);
	double tmp;
	if (fabs(t) <= 3.8e-160) {
		tmp = ((cos(k) * l) * (l / (k * k))) * (2.0 / (pow(k, 2.0) * fabs(t)));
	} else if (fabs(t) <= 6e-25) {
		tmp = ((2.0 / t_2) * l) * (l / ((fabs(t) * fma(k, (k / t_2), 2.0)) * pow(k, 2.0)));
	} else {
		tmp = (l / ((t_1 * fabs(t)) * t_1)) * l;
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k * abs(t))
	t_2 = Float64(abs(t) * abs(t))
	tmp = 0.0
	if (abs(t) <= 3.8e-160)
		tmp = Float64(Float64(Float64(cos(k) * l) * Float64(l / Float64(k * k))) * Float64(2.0 / Float64((k ^ 2.0) * abs(t))));
	elseif (abs(t) <= 6e-25)
		tmp = Float64(Float64(Float64(2.0 / t_2) * l) * Float64(l / Float64(Float64(abs(t) * fma(k, Float64(k / t_2), 2.0)) * (k ^ 2.0))));
	else
		tmp = Float64(Float64(l / Float64(Float64(t_1 * abs(t)) * t_1)) * l);
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 3.8e-160], N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 6e-25], N[(N[(N[(2.0 / t$95$2), $MachinePrecision] * l), $MachinePrecision] * N[(l / N[(N[(N[Abs[t], $MachinePrecision] * N[(k * N[(k / t$95$2), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.7999999999999998e-160

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 60.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 60.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot t}}{\color{blue}{{\sin k}^{2}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot t}}{\color{blue}{{\sin k}^{2}}} \]

   6. Applied rewrites 57.5%

      \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{0.5 - 0.5 \cdot \cos \left(k + k\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      5. times-frac N/A

         \[\leadsto \frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \frac{2}{t}}{\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\frac{2}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\frac{2}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \frac{\frac{\color{blue}{2}}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      9. associate-/l* N/A

         \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\color{blue}{\frac{2}{t}}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\color{blue}{\frac{2}{t}}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{\color{blue}{t}}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      13. lower-/.f64 60.4%

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\color{blue}{0.5} - 0.5 \cdot \cos \left(k + k\right)} \]

      14. lift--.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(k + k\right)}} \]

      16. fp-cancel-sub-sign-inv N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(k + k\right)}} \]

   8. Applied rewrites 60.4%

      \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \color{blue}{\frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{2}{\color{blue}{{k}^{2} \cdot t}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{2}{{k}^{2} \cdot \color{blue}{t}} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{2}{{k}^{2} \cdot t} \]

       3. lower-pow.f64 57.8%

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{2}{{k}^{2} \cdot t} \]

   11. Applied rewrites 57.8%

       \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{2}{\color{blue}{{k}^{2} \cdot t}} \]

   if 3.7999999999999998e-160 < t < 5.9999999999999995e-25

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      4. mult-flip N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \cdot \frac{1}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \frac{1}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k} \cdot \frac{1}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{1}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\sin k \cdot \tan k}} \cdot \frac{1}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. frac-times N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot 1}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot 1}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   3. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{\left(\frac{2}{\left(t \cdot t\right) \cdot t} \cdot \left(\ell \cdot \ell\right)\right) \cdot 1}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{2}{\left(t \cdot t\right) \cdot t} \cdot \left(\ell \cdot \ell\right)\right) \cdot 1}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      2. *-rgt-identity 50.6%

         \[\leadsto \frac{\color{blue}{\frac{2}{\left(t \cdot t\right) \cdot t} \cdot \left(\ell \cdot \ell\right)}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2}{\left(t \cdot t\right) \cdot t} \cdot \left(\ell \cdot \ell\right)}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2}{\left(t \cdot t\right) \cdot t}} \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      6. associate-/r* N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t \cdot t}}{t}} \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t \cdot t} \cdot \left(\ell \cdot \ell\right)}{t}}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t \cdot t} \cdot \left(\ell \cdot \ell\right)}{t}}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{2}{t \cdot t} \cdot \left(\ell \cdot \ell\right)}}{t}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      10. lower-/.f64 54.1%

          \[\leadsto \frac{\frac{\color{blue}{\frac{2}{t \cdot t}} \cdot \left(\ell \cdot \ell\right)}{t}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

   5. Applied rewrites 54.1%

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t \cdot t} \cdot \left(\ell \cdot \ell\right)}{t}}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t \cdot t} \cdot \left(\ell \cdot \ell\right)}{t}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t \cdot t} \cdot \left(\ell \cdot \ell\right)}{t}}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{t \cdot t} \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2}{t \cdot t} \cdot \left(\ell \cdot \ell\right)}}{t \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{t \cdot t} \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{t \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \ell}}{t \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{2}{t \cdot t} \cdot \ell\right)} \cdot \frac{\ell}{t \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \color{blue}{\frac{\ell}{t \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)}} \]

      12. *-commutative N/A

          \[\leadsto \left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \frac{\ell}{t \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\tan k \cdot \sin k\right)\right)}} \]

   7. Applied rewrites 54.4%

      \[\leadsto \color{blue}{\left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \frac{\ell}{\left(t \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \left(\tan k \cdot \sin k\right)}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \frac{\ell}{\left(t \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \color{blue}{{k}^{2}}} \]
   9. Step-by-step derivation

      1. lower-pow.f64 49.0%

         \[\leadsto \left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \frac{\ell}{\left(t \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot {k}^{\color{blue}{2}}} \]

   10. Applied rewrites 49.0%

       \[\leadsto \left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \frac{\ell}{\left(t \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \color{blue}{{k}^{2}}} \]

   if 5.9999999999999995e-25 < t

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.8%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.2%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. pow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      12. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{{k}^{2}}} \]

      13. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

      14. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      15. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      17. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.5%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 59.5%

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      13. lower-*.f64 62.6%

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lower-*.f64 66.2%

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 66.2%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 72.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_1 := \left|t\right| \cdot \left|t\right|\\ t_2 := k \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 3.8 \cdot 10^{-160}:\\ \;\;\;\;\left(\left(1 \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{\left|t\right|}}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}\\ \mathbf{elif}\;\left|t\right| \leq 6 \cdot 10^{-25}:\\ \;\;\;\;\left(\frac{2}{t\_1} \cdot \ell\right) \cdot \frac{\ell}{\left(\left|t\right| \cdot \mathsf{fma}\left(k, \frac{k}{t\_1}, 2\right)\right) \cdot {k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t\_2 \cdot \left|t\right|\right) \cdot t\_2} \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (fabs t) (fabs t))) (t_2 (* k (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 3.8e-160)
      (*
       (* (* 1.0 l) (/ l (* k k)))
       (/ (/ 2.0 (fabs t)) (fma (cos (+ k k)) -0.5 0.5)))
      (if (<= (fabs t) 6e-25)
        (*
         (* (/ 2.0 t_1) l)
         (/ l (* (* (fabs t) (fma k (/ k t_1) 2.0)) (pow k 2.0))))
        (* (/ l (* (* t_2 (fabs t)) t_2)) l))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) * fabs(t);
	double t_2 = k * fabs(t);
	double tmp;
	if (fabs(t) <= 3.8e-160) {
		tmp = ((1.0 * l) * (l / (k * k))) * ((2.0 / fabs(t)) / fma(cos((k + k)), -0.5, 0.5));
	} else if (fabs(t) <= 6e-25) {
		tmp = ((2.0 / t_1) * l) * (l / ((fabs(t) * fma(k, (k / t_1), 2.0)) * pow(k, 2.0)));
	} else {
		tmp = (l / ((t_2 * fabs(t)) * t_2)) * l;
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) * abs(t))
	t_2 = Float64(k * abs(t))
	tmp = 0.0
	if (abs(t) <= 3.8e-160)
		tmp = Float64(Float64(Float64(1.0 * l) * Float64(l / Float64(k * k))) * Float64(Float64(2.0 / abs(t)) / fma(cos(Float64(k + k)), -0.5, 0.5)));
	elseif (abs(t) <= 6e-25)
		tmp = Float64(Float64(Float64(2.0 / t_1) * l) * Float64(l / Float64(Float64(abs(t) * fma(k, Float64(k / t_1), 2.0)) * (k ^ 2.0))));
	else
		tmp = Float64(Float64(l / Float64(Float64(t_2 * abs(t)) * t_2)) * l);
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 3.8e-160], N[(N[(N[(1.0 * l), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 6e-25], N[(N[(N[(2.0 / t$95$1), $MachinePrecision] * l), $MachinePrecision] * N[(l / N[(N[(N[Abs[t], $MachinePrecision] * N[(k * N[(k / t$95$1), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(t$95$2 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.7999999999999998e-160

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 60.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 60.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{{\sin k}^{2}}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot t}}{\color{blue}{{\sin k}^{2}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot t}}{\color{blue}{{\sin k}^{2}}} \]

   6. Applied rewrites 57.5%

      \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{0.5 - 0.5 \cdot \cos \left(k + k\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      5. times-frac N/A

         \[\leadsto \frac{\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \frac{2}{t}}{\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\frac{2}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\frac{2}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \frac{\frac{\color{blue}{2}}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      9. associate-/l* N/A

         \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\color{blue}{\frac{2}{t}}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\color{blue}{\frac{2}{t}}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{\color{blue}{t}}}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      13. lower-/.f64 60.4%

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\color{blue}{0.5} - 0.5 \cdot \cos \left(k + k\right)} \]

      14. lift--.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(k + k\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(k + k\right)}} \]

      16. fp-cancel-sub-sign-inv N/A

          \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(k + k\right)}} \]

   8. Applied rewrites 60.4%

      \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \color{blue}{\frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \left(\left(1 \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 53.1%

       \[\leadsto \left(\left(1 \cdot \ell\right) \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\frac{2}{t}}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)} \]

   if 3.7999999999999998e-160 < t < 5.9999999999999995e-25

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      4. mult-flip N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \cdot \frac{1}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \frac{1}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k} \cdot \frac{1}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{1}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\sin k \cdot \tan k}} \cdot \frac{1}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. frac-times N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot 1}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot 1}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   3. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{\left(\frac{2}{\left(t \cdot t\right) \cdot t} \cdot \left(\ell \cdot \ell\right)\right) \cdot 1}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{2}{\left(t \cdot t\right) \cdot t} \cdot \left(\ell \cdot \ell\right)\right) \cdot 1}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      2. *-rgt-identity 50.6%

         \[\leadsto \frac{\color{blue}{\frac{2}{\left(t \cdot t\right) \cdot t} \cdot \left(\ell \cdot \ell\right)}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2}{\left(t \cdot t\right) \cdot t} \cdot \left(\ell \cdot \ell\right)}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2}{\left(t \cdot t\right) \cdot t}} \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      6. associate-/r* N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t \cdot t}}{t}} \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t \cdot t} \cdot \left(\ell \cdot \ell\right)}{t}}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t \cdot t} \cdot \left(\ell \cdot \ell\right)}{t}}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{2}{t \cdot t} \cdot \left(\ell \cdot \ell\right)}}{t}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      10. lower-/.f64 54.1%

          \[\leadsto \frac{\frac{\color{blue}{\frac{2}{t \cdot t}} \cdot \left(\ell \cdot \ell\right)}{t}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

   5. Applied rewrites 54.1%

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t \cdot t} \cdot \left(\ell \cdot \ell\right)}{t}}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t \cdot t} \cdot \left(\ell \cdot \ell\right)}{t}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t \cdot t} \cdot \left(\ell \cdot \ell\right)}{t}}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{t \cdot t} \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2}{t \cdot t} \cdot \left(\ell \cdot \ell\right)}}{t \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{t \cdot t} \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{t \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \ell}}{t \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{2}{t \cdot t} \cdot \ell\right)} \cdot \frac{\ell}{t \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \color{blue}{\frac{\ell}{t \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)}} \]

      12. *-commutative N/A

          \[\leadsto \left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \frac{\ell}{t \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\tan k \cdot \sin k\right)\right)}} \]

   7. Applied rewrites 54.4%

      \[\leadsto \color{blue}{\left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \frac{\ell}{\left(t \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \left(\tan k \cdot \sin k\right)}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \frac{\ell}{\left(t \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \color{blue}{{k}^{2}}} \]
   9. Step-by-step derivation

      1. lower-pow.f64 49.0%

         \[\leadsto \left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \frac{\ell}{\left(t \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot {k}^{\color{blue}{2}}} \]

   10. Applied rewrites 49.0%

       \[\leadsto \left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \frac{\ell}{\left(t \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \color{blue}{{k}^{2}}} \]

   if 5.9999999999999995e-25 < t

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.8%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.2%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. pow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      12. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{{k}^{2}}} \]

      13. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

      14. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      15. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      17. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.5%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 59.5%

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      13. lower-*.f64 62.6%

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lower-*.f64 66.2%

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 66.2%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 70.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_1 := k \cdot \left|t\right|\\ t_2 := \left|t\right| \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 3.8 \cdot 10^{-160}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \left|t\right|}\\ \mathbf{elif}\;\left|t\right| \leq 6 \cdot 10^{-25}:\\ \;\;\;\;\left(\frac{2}{t\_2} \cdot \ell\right) \cdot \frac{\ell}{\left(\left|t\right| \cdot \mathsf{fma}\left(k, \frac{k}{t\_2}, 2\right)\right) \cdot {k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot \left|t\right|\right) \cdot t\_1} \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (fabs t))) (t_2 (* (fabs t) (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 3.8e-160)
      (* 2.0 (/ (pow l 2.0) (* (pow k 4.0) (fabs t))))
      (if (<= (fabs t) 6e-25)
        (*
         (* (/ 2.0 t_2) l)
         (/ l (* (* (fabs t) (fma k (/ k t_2) 2.0)) (pow k 2.0))))
        (* (/ l (* (* t_1 (fabs t)) t_1)) l))))))

C

double code(double t, double l, double k) {
	double t_1 = k * fabs(t);
	double t_2 = fabs(t) * fabs(t);
	double tmp;
	if (fabs(t) <= 3.8e-160) {
		tmp = 2.0 * (pow(l, 2.0) / (pow(k, 4.0) * fabs(t)));
	} else if (fabs(t) <= 6e-25) {
		tmp = ((2.0 / t_2) * l) * (l / ((fabs(t) * fma(k, (k / t_2), 2.0)) * pow(k, 2.0)));
	} else {
		tmp = (l / ((t_1 * fabs(t)) * t_1)) * l;
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k * abs(t))
	t_2 = Float64(abs(t) * abs(t))
	tmp = 0.0
	if (abs(t) <= 3.8e-160)
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64((k ^ 4.0) * abs(t))));
	elseif (abs(t) <= 6e-25)
		tmp = Float64(Float64(Float64(2.0 / t_2) * l) * Float64(l / Float64(Float64(abs(t) * fma(k, Float64(k / t_2), 2.0)) * (k ^ 2.0))));
	else
		tmp = Float64(Float64(l / Float64(Float64(t_1 * abs(t)) * t_1)) * l);
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 3.8e-160], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[k, 4.0], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 6e-25], N[(N[(N[(2.0 / t$95$2), $MachinePrecision] * l), $MachinePrecision] * N[(l / N[(N[(N[Abs[t], $MachinePrecision] * N[(k * N[(k / t$95$2), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.7999999999999998e-160

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 60.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 60.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

      2. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

      4. lower-pow.f64 51.8%

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

   7. Applied rewrites 51.8%

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   if 3.7999999999999998e-160 < t < 5.9999999999999995e-25

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      4. mult-flip N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \cdot \frac{1}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \cdot \frac{1}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k} \cdot \frac{1}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{1}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\sin k \cdot \tan k}} \cdot \frac{1}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. frac-times N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot 1}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot 1}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   3. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{\left(\frac{2}{\left(t \cdot t\right) \cdot t} \cdot \left(\ell \cdot \ell\right)\right) \cdot 1}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{2}{\left(t \cdot t\right) \cdot t} \cdot \left(\ell \cdot \ell\right)\right) \cdot 1}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      2. *-rgt-identity 50.6%

         \[\leadsto \frac{\color{blue}{\frac{2}{\left(t \cdot t\right) \cdot t} \cdot \left(\ell \cdot \ell\right)}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2}{\left(t \cdot t\right) \cdot t} \cdot \left(\ell \cdot \ell\right)}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2}{\left(t \cdot t\right) \cdot t}} \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      6. associate-/r* N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t \cdot t}}{t}} \cdot \left(\ell \cdot \ell\right)}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t \cdot t} \cdot \left(\ell \cdot \ell\right)}{t}}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t \cdot t} \cdot \left(\ell \cdot \ell\right)}{t}}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{2}{t \cdot t} \cdot \left(\ell \cdot \ell\right)}}{t}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      10. lower-/.f64 54.1%

          \[\leadsto \frac{\frac{\color{blue}{\frac{2}{t \cdot t}} \cdot \left(\ell \cdot \ell\right)}{t}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

   5. Applied rewrites 54.1%

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t \cdot t} \cdot \left(\ell \cdot \ell\right)}{t}}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t \cdot t} \cdot \left(\ell \cdot \ell\right)}{t}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t \cdot t} \cdot \left(\ell \cdot \ell\right)}{t}}}{\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{t \cdot t} \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2}{t \cdot t} \cdot \left(\ell \cdot \ell\right)}}{t \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{t \cdot t} \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{t \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \ell}}{t \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{2}{t \cdot t} \cdot \ell\right)} \cdot \frac{\ell}{t \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \color{blue}{\frac{\ell}{t \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(\tan k \cdot \sin k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)\right)}} \]

      12. *-commutative N/A

          \[\leadsto \left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \frac{\ell}{t \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\tan k \cdot \sin k\right)\right)}} \]

   7. Applied rewrites 54.4%

      \[\leadsto \color{blue}{\left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \frac{\ell}{\left(t \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \left(\tan k \cdot \sin k\right)}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \frac{\ell}{\left(t \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \color{blue}{{k}^{2}}} \]
   9. Step-by-step derivation

      1. lower-pow.f64 49.0%

         \[\leadsto \left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \frac{\ell}{\left(t \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot {k}^{\color{blue}{2}}} \]

   10. Applied rewrites 49.0%

       \[\leadsto \left(\frac{2}{t \cdot t} \cdot \ell\right) \cdot \frac{\ell}{\left(t \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \color{blue}{{k}^{2}}} \]

   if 5.9999999999999995e-25 < t

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.8%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.2%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. pow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      12. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{{k}^{2}}} \]

      13. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

      14. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      15. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      17. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.5%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 59.5%

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      13. lower-*.f64 62.6%

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lower-*.f64 66.2%

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 66.2%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 68.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_1 := k \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.04 \cdot 10^{-109}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \left|t\right|}\\ \mathbf{elif}\;\left|t\right| \leq 5.5 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{\left|t\right|}}{\left|t\right| \cdot \left|t\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot \left|t\right|\right) \cdot t\_1} \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1.04e-109)
      (* 2.0 (/ (pow l 2.0) (* (pow k 4.0) (fabs t))))
      (if (<= (fabs t) 5.5e+69)
        (/ (/ (* l (/ l (* k k))) (fabs t)) (* (fabs t) (fabs t)))
        (* (/ l (* (* t_1 (fabs t)) t_1)) l))))))

C

double code(double t, double l, double k) {
	double t_1 = k * fabs(t);
	double tmp;
	if (fabs(t) <= 1.04e-109) {
		tmp = 2.0 * (pow(l, 2.0) / (pow(k, 4.0) * fabs(t)));
	} else if (fabs(t) <= 5.5e+69) {
		tmp = ((l * (l / (k * k))) / fabs(t)) / (fabs(t) * fabs(t));
	} else {
		tmp = (l / ((t_1 * fabs(t)) * t_1)) * l;
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = k * Math.abs(t);
	double tmp;
	if (Math.abs(t) <= 1.04e-109) {
		tmp = 2.0 * (Math.pow(l, 2.0) / (Math.pow(k, 4.0) * Math.abs(t)));
	} else if (Math.abs(t) <= 5.5e+69) {
		tmp = ((l * (l / (k * k))) / Math.abs(t)) / (Math.abs(t) * Math.abs(t));
	} else {
		tmp = (l / ((t_1 * Math.abs(t)) * t_1)) * l;
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = k * math.fabs(t)
	tmp = 0
	if math.fabs(t) <= 1.04e-109:
		tmp = 2.0 * (math.pow(l, 2.0) / (math.pow(k, 4.0) * math.fabs(t)))
	elif math.fabs(t) <= 5.5e+69:
		tmp = ((l * (l / (k * k))) / math.fabs(t)) / (math.fabs(t) * math.fabs(t))
	else:
		tmp = (l / ((t_1 * math.fabs(t)) * t_1)) * l
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * abs(t))
	tmp = 0.0
	if (abs(t) <= 1.04e-109)
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64((k ^ 4.0) * abs(t))));
	elseif (abs(t) <= 5.5e+69)
		tmp = Float64(Float64(Float64(l * Float64(l / Float64(k * k))) / abs(t)) / Float64(abs(t) * abs(t)));
	else
		tmp = Float64(Float64(l / Float64(Float64(t_1 * abs(t)) * t_1)) * l);
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * abs(t);
	tmp = 0.0;
	if (abs(t) <= 1.04e-109)
		tmp = 2.0 * ((l ^ 2.0) / ((k ^ 4.0) * abs(t)));
	elseif (abs(t) <= 5.5e+69)
		tmp = ((l * (l / (k * k))) / abs(t)) / (abs(t) * abs(t));
	else
		tmp = (l / ((t_1 * abs(t)) * t_1)) * l;
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.04e-109], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[k, 4.0], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 5.5e+69], N[(N[(N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.04e-109

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 60.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 60.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

      2. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

      4. lower-pow.f64 51.8%

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

   7. Applied rewrites 51.8%

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   if 1.04e-109 < t < 5.5e69

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.8%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      7. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      10. associate-/r* N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{\color{blue}{\left(t \cdot t\right) \cdot t}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{\left(t \cdot t\right) \cdot \color{blue}{t}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t}}{\color{blue}{t \cdot t}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t}}{\color{blue}{t \cdot t}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t}}{\color{blue}{t} \cdot t} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t}}{t \cdot t} \]

      17. associate-/l* N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t}}{t \cdot t} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t}}{t \cdot t} \]

      19. lower-/.f64 58.0%

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t}}{t \cdot t} \]

      20. lift-pow.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t}}{t \cdot t} \]

      21. unpow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t}}{t \cdot t} \]

      22. lower-*.f64 58.0%

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t}}{t \cdot t} \]

   6. Applied rewrites 58.0%

      \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t}}{\color{blue}{t \cdot t}} \]

   if 5.5e69 < t

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.8%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.2%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. pow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      12. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{{k}^{2}}} \]

      13. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

      14. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      15. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      17. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.5%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 59.5%

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      13. lower-*.f64 62.6%

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lower-*.f64 66.2%

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 66.2%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 66.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} t_1 := k \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 5.5 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{\left|t\right|}}{\left|t\right| \cdot \left|t\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot \left|t\right|\right) \cdot t\_1} \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 5.5e+69)
      (/ (/ (* l (/ l (* k k))) (fabs t)) (* (fabs t) (fabs t)))
      (* (/ l (* (* t_1 (fabs t)) t_1)) l)))))

C

double code(double t, double l, double k) {
	double t_1 = k * fabs(t);
	double tmp;
	if (fabs(t) <= 5.5e+69) {
		tmp = ((l * (l / (k * k))) / fabs(t)) / (fabs(t) * fabs(t));
	} else {
		tmp = (l / ((t_1 * fabs(t)) * t_1)) * l;
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = k * Math.abs(t);
	double tmp;
	if (Math.abs(t) <= 5.5e+69) {
		tmp = ((l * (l / (k * k))) / Math.abs(t)) / (Math.abs(t) * Math.abs(t));
	} else {
		tmp = (l / ((t_1 * Math.abs(t)) * t_1)) * l;
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = k * math.fabs(t)
	tmp = 0
	if math.fabs(t) <= 5.5e+69:
		tmp = ((l * (l / (k * k))) / math.fabs(t)) / (math.fabs(t) * math.fabs(t))
	else:
		tmp = (l / ((t_1 * math.fabs(t)) * t_1)) * l
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * abs(t))
	tmp = 0.0
	if (abs(t) <= 5.5e+69)
		tmp = Float64(Float64(Float64(l * Float64(l / Float64(k * k))) / abs(t)) / Float64(abs(t) * abs(t)));
	else
		tmp = Float64(Float64(l / Float64(Float64(t_1 * abs(t)) * t_1)) * l);
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * abs(t);
	tmp = 0.0;
	if (abs(t) <= 5.5e+69)
		tmp = ((l * (l / (k * k))) / abs(t)) / (abs(t) * abs(t));
	else
		tmp = (l / ((t_1 * abs(t)) * t_1)) * l;
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 5.5e+69], N[(N[(N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.5e69

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.8%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      7. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      10. associate-/r* N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{\color{blue}{\left(t \cdot t\right) \cdot t}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{\left(t \cdot t\right) \cdot \color{blue}{t}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t}}{\color{blue}{t \cdot t}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t}}{\color{blue}{t \cdot t}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t}}{\color{blue}{t} \cdot t} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t}}{t \cdot t} \]

      17. associate-/l* N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t}}{t \cdot t} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t}}{t \cdot t} \]

      19. lower-/.f64 58.0%

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t}}{t \cdot t} \]

      20. lift-pow.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t}}{t \cdot t} \]

      21. unpow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t}}{t \cdot t} \]

      22. lower-*.f64 58.0%

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t}}{t \cdot t} \]

   6. Applied rewrites 58.0%

      \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t}}{\color{blue}{t \cdot t}} \]

   if 5.5e69 < t

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.8%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.2%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. pow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      12. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{{k}^{2}}} \]

      13. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

      14. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      15. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      17. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.5%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 59.5%

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      13. lower-*.f64 62.6%

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lower-*.f64 66.2%

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 66.2%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 66.2% accurate, 6.6× speedup

Math

\[\frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ l (* (* (* k t) t) (* k t))) l))

C

double code(double t, double l, double k) {
	return (l / (((k * t) * t) * (k * t))) * l;
}

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 = (l / (((k * t) * t) * (k * t))) * l
end function

Java

public static double code(double t, double l, double k) {
	return (l / (((k * t) * t) * (k * t))) * l;
}

Python

def code(t, l, k):
	return (l / (((k * t) * t) * (k * t))) * l

Julia

function code(t, l, k)
	return Float64(Float64(l / Float64(Float64(Float64(k * t) * t) * Float64(k * t))) * l)
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / (((k * t) * t) * (k * t))) * l;
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[(N[(N[(k * t), $MachinePrecision] * t), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]

Derivation

1. Initial program 54.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   4. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   5. lower-pow.f64 50.8%

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 50.8%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   6. lower-/.f64 55.2%

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   8. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   9. pow3 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   12. *-commutative N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{{k}^{2}}} \]

   13. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

   14. unpow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

   15. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

   16. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

   17. lower-*.f64 59.5%

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

6. Applied rewrites 59.5%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   3. lower-*.f64 59.5%

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   5. *-commutative N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   8. associate-*l* N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

   9. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   10. *-commutative N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   13. lower-*.f64 62.6%

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

8. Applied rewrites 62.6%

   \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   3. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   5. lower-*.f64 66.2%

      \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

10. Applied rewrites 66.2%

    \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
11. Add Preprocessing

Alternative 17: 62.6% accurate, 6.6× speedup

Math

\[\frac{\ell}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ l (* (* (* k t) k) (* t t))) l))

C

double code(double t, double l, double k) {
	return (l / (((k * t) * k) * (t * t))) * l;
}

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 = (l / (((k * t) * k) * (t * t))) * l
end function

Java

public static double code(double t, double l, double k) {
	return (l / (((k * t) * k) * (t * t))) * l;
}

Python

def code(t, l, k):
	return (l / (((k * t) * k) * (t * t))) * l

Julia

function code(t, l, k)
	return Float64(Float64(l / Float64(Float64(Float64(k * t) * k) * Float64(t * t))) * l)
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / (((k * t) * k) * (t * t))) * l;
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[(N[(N[(k * t), $MachinePrecision] * k), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]

Derivation

1. Initial program 54.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   4. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   5. lower-pow.f64 50.8%

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 50.8%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   6. lower-/.f64 55.2%

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   8. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   9. pow3 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   12. *-commutative N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{{k}^{2}}} \]

   13. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

   14. unpow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

   15. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

   16. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

   17. lower-*.f64 59.5%

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

6. Applied rewrites 59.5%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   3. lower-*.f64 59.5%

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   5. *-commutative N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   8. associate-*l* N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

   9. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   10. *-commutative N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   13. lower-*.f64 62.6%

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

8. Applied rewrites 62.6%

   \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

   4. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

   6. lower-*.f64 61.2%

      \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

10. Applied rewrites 61.2%

    \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \left(t \cdot t\right)} \cdot \ell \]
11. Add Preprocessing

Alternative 18: 61.2% accurate, 6.6× speedup

Math

\[\frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ l (* (* k (* t t)) (* k t))) l))

C

double code(double t, double l, double k) {
	return (l / ((k * (t * t)) * (k * t))) * l;
}

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 = (l / ((k * (t * t)) * (k * t))) * l
end function

Java

public static double code(double t, double l, double k) {
	return (l / ((k * (t * t)) * (k * t))) * l;
}

Python

def code(t, l, k):
	return (l / ((k * (t * t)) * (k * t))) * l

Julia

function code(t, l, k)
	return Float64(Float64(l / Float64(Float64(k * Float64(t * t)) * Float64(k * t))) * l)
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / ((k * (t * t)) * (k * t))) * l;
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[(N[(k * N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]

Derivation

1. Initial program 54.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   4. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   5. lower-pow.f64 50.8%

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 50.8%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   6. lower-/.f64 55.2%

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   8. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   9. pow3 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   12. *-commutative N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{{k}^{2}}} \]

   13. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

   14. unpow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

   15. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

   16. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

   17. lower-*.f64 59.5%

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

6. Applied rewrites 59.5%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   3. lower-*.f64 59.5%

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   5. *-commutative N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   8. associate-*l* N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

   9. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   10. *-commutative N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   13. lower-*.f64 62.6%

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

8. Applied rewrites 62.6%

   \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025196 
(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))))