Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.1% → 91.5%

Time: 7.2s

Alternatives: 18

Speedup: 6.6×

• 27.3% 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: 54.1% 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: 91.5% accurate, 1.1× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.66 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{\cos k}{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot k\right) \cdot \left|t\right|} \cdot \left(\ell + \ell\right)\right) \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left|t\right|}{\ell} \cdot \left(\left(\frac{\sin k \cdot \left|t\right|}{\ell} \cdot \left|t\right|\right) \cdot \left(\mathsf{fma}\left(\frac{k}{\left|t\right| \cdot \left|t\right|}, k, 2\right) \cdot \tan k\right)\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 1.66e-6)
    (*
     (* (/ (cos k) (* (* (fma (cos (+ k k)) -0.5 0.5) k) (fabs t))) (+ l l))
     (/ l k))
    (/
     2.0
     (*
      (/ (fabs t) l)
      (*
       (* (/ (* (sin k) (fabs t)) l) (fabs t))
       (* (fma (/ k (* (fabs t) (fabs t))) k 2.0) (tan k))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 1.66e-6) {
		tmp = ((cos(k) / ((fma(cos((k + k)), -0.5, 0.5) * k) * fabs(t))) * (l + l)) * (l / k);
	} else {
		tmp = 2.0 / ((fabs(t) / l) * ((((sin(k) * fabs(t)) / l) * fabs(t)) * (fma((k / (fabs(t) * fabs(t))), k, 2.0) * tan(k))));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 1.66e-6)
		tmp = Float64(Float64(Float64(cos(k) / Float64(Float64(fma(cos(Float64(k + k)), -0.5, 0.5) * k) * abs(t))) * Float64(l + l)) * Float64(l / k));
	else
		tmp = Float64(2.0 / Float64(Float64(abs(t) / l) * Float64(Float64(Float64(Float64(sin(k) * abs(t)) / l) * abs(t)) * Float64(fma(Float64(k / Float64(abs(t) * abs(t))), k, 2.0) * tan(k)))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.66e-6], N[(N[(N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * k), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.65999999999999999e-6

   1. Initial program 54.1%

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

   2. 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 59.7

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

   4. Applied rewrites 59.7%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 59.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 59.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 59.8

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

   8. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   10. Applied rewrites 69.1%

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

   if 1.65999999999999999e-6 < t

   1. Initial program 54.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.7

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

   3. Applied rewrites 66.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.9

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

   5. Applied rewrites 75.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 79.2

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

   7. Applied rewrites 69.7%

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

Alternative 2: 90.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \sin k \cdot \left|t\right|\\ t_2 := \frac{\left|t\right|}{\ell}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.66 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{\cos k}{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot k\right) \cdot \left|t\right|} \cdot \left(\ell + \ell\right)\right) \cdot \frac{\ell}{k}\\ \mathbf{elif}\;\left|t\right| \leq 5.2 \cdot 10^{+138}:\\ \;\;\;\;\frac{2}{\left(\frac{\tan k}{\ell} \cdot \mathsf{fma}\left(k, \frac{k}{\left|t\right| \cdot \left|t\right|}, 2\right)\right) \cdot \left(\left(t\_1 \cdot \left|t\right|\right) \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t\_2 \cdot \left(\left|t\right| \cdot \frac{t\_1}{\ell}\right)\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (fabs t))) (t_2 (/ (fabs t) l)))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1.66e-6)
      (*
       (* (/ (cos k) (* (* (fma (cos (+ k k)) -0.5 0.5) k) (fabs t))) (+ l l))
       (/ l k))
      (if (<= (fabs t) 5.2e+138)
        (/
         2.0
         (*
          (* (/ (tan k) l) (fma k (/ k (* (fabs t) (fabs t))) 2.0))
          (* (* t_1 (fabs t)) t_2)))
        (/ 2.0 (* (* (* t_2 (* (fabs t) (/ t_1 l))) (tan k)) 2.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * fabs(t);
	double t_2 = fabs(t) / l;
	double tmp;
	if (fabs(t) <= 1.66e-6) {
		tmp = ((cos(k) / ((fma(cos((k + k)), -0.5, 0.5) * k) * fabs(t))) * (l + l)) * (l / k);
	} else if (fabs(t) <= 5.2e+138) {
		tmp = 2.0 / (((tan(k) / l) * fma(k, (k / (fabs(t) * fabs(t))), 2.0)) * ((t_1 * fabs(t)) * t_2));
	} else {
		tmp = 2.0 / (((t_2 * (fabs(t) * (t_1 / l))) * tan(k)) * 2.0);
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * abs(t))
	t_2 = Float64(abs(t) / l)
	tmp = 0.0
	if (abs(t) <= 1.66e-6)
		tmp = Float64(Float64(Float64(cos(k) / Float64(Float64(fma(cos(Float64(k + k)), -0.5, 0.5) * k) * abs(t))) * Float64(l + l)) * Float64(l / k));
	elseif (abs(t) <= 5.2e+138)
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) / l) * fma(k, Float64(k / Float64(abs(t) * abs(t))), 2.0)) * Float64(Float64(t_1 * abs(t)) * t_2)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * Float64(abs(t) * Float64(t_1 / l))) * tan(k)) * 2.0));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.66e-6], N[(N[(N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * k), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 5.2e+138], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(k * N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$2 * N[(N[Abs[t], $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.65999999999999999e-6

   1. Initial program 54.1%

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

   2. 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 59.7

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

   4. Applied rewrites 59.7%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 59.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 59.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 59.8

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

   8. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   10. Applied rewrites 69.1%

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

   if 1.65999999999999999e-6 < t < 5.2000000000000002e138

   1. Initial program 54.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.7

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

   3. Applied rewrites 66.7%

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

   5. Applied rewrites 60.0%

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

   if 5.2000000000000002e138 < t

   1. Initial program 54.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.7

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

   3. Applied rewrites 66.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.9

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

   5. Applied rewrites 75.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 75.9

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

   7. Applied rewrites 75.9%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 68.3%

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

Alternative 3: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \sin k \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.66 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{\cos k}{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot k\right) \cdot \left|t\right|} \cdot \left(\ell + \ell\right)\right) \cdot \frac{\ell}{k}\\ \mathbf{elif}\;\left|t\right| \leq 8 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{2}{\left(t\_1 \cdot \left|t\right|\right) \cdot \left|t\right|} \cdot \ell}{\mathsf{fma}\left(k, \frac{k}{\left|t\right| \cdot \left|t\right|}, 2\right) \cdot \tan k} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left|t\right|}{\ell} \cdot \left(\left|t\right| \cdot \frac{t\_1}{\ell}\right)\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1.66e-6)
      (*
       (* (/ (cos k) (* (* (fma (cos (+ k k)) -0.5 0.5) k) (fabs t))) (+ l l))
       (/ l k))
      (if (<= (fabs t) 8e+102)
        (*
         (/
          (* (/ 2.0 (* (* t_1 (fabs t)) (fabs t))) l)
          (* (fma k (/ k (* (fabs t) (fabs t))) 2.0) (tan k)))
         l)
        (/
         2.0
         (* (* (* (/ (fabs t) l) (* (fabs t) (/ t_1 l))) (tan k)) 2.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * fabs(t);
	double tmp;
	if (fabs(t) <= 1.66e-6) {
		tmp = ((cos(k) / ((fma(cos((k + k)), -0.5, 0.5) * k) * fabs(t))) * (l + l)) * (l / k);
	} else if (fabs(t) <= 8e+102) {
		tmp = (((2.0 / ((t_1 * fabs(t)) * fabs(t))) * l) / (fma(k, (k / (fabs(t) * fabs(t))), 2.0) * tan(k))) * l;
	} else {
		tmp = 2.0 / ((((fabs(t) / l) * (fabs(t) * (t_1 / l))) * tan(k)) * 2.0);
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * abs(t))
	tmp = 0.0
	if (abs(t) <= 1.66e-6)
		tmp = Float64(Float64(Float64(cos(k) / Float64(Float64(fma(cos(Float64(k + k)), -0.5, 0.5) * k) * abs(t))) * Float64(l + l)) * Float64(l / k));
	elseif (abs(t) <= 8e+102)
		tmp = Float64(Float64(Float64(Float64(2.0 / Float64(Float64(t_1 * abs(t)) * abs(t))) * l) / Float64(fma(k, Float64(k / Float64(abs(t) * abs(t))), 2.0) * tan(k))) * l);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(abs(t) / l) * Float64(abs(t) * Float64(t_1 / l))) * tan(k)) * 2.0));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $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], 1.66e-6], N[(N[(N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * k), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 8e+102], N[(N[(N[(N[(2.0 / N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / N[(N[(k * N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.65999999999999999e-6

   1. Initial program 54.1%

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

   2. 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 59.7

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

   4. Applied rewrites 59.7%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 59.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 59.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 59.8

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

   8. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   10. Applied rewrites 69.1%

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

   if 1.65999999999999999e-6 < t < 7.99999999999999982e102

   1. Initial program 54.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.7

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

   3. Applied rewrites 66.7%

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

   5. Applied rewrites 56.3%

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

   if 7.99999999999999982e102 < t

   1. Initial program 54.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.7

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

   3. Applied rewrites 66.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.9

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

   5. Applied rewrites 75.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 75.9

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

   7. Applied rewrites 75.9%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 68.3%

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

Alternative 4: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \sin k \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.66 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{\cos k}{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot k\right) \cdot \left|t\right|} \cdot \left(\ell + \ell\right)\right) \cdot \frac{\ell}{k}\\ \mathbf{elif}\;\left|t\right| \leq 8 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\left(\frac{\tan k}{\ell} \cdot \mathsf{fma}\left(k, \frac{k}{\left|t\right| \cdot \left|t\right|}, 2\right)\right) \cdot \left(\left(t\_1 \cdot \left|t\right|\right) \cdot \left|t\right|\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left|t\right|}{\ell} \cdot \left(\left|t\right| \cdot \frac{t\_1}{\ell}\right)\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1.66e-6)
      (*
       (* (/ (cos k) (* (* (fma (cos (+ k k)) -0.5 0.5) k) (fabs t))) (+ l l))
       (/ l k))
      (if (<= (fabs t) 8e+102)
        (*
         (/
          2.0
          (*
           (* (/ (tan k) l) (fma k (/ k (* (fabs t) (fabs t))) 2.0))
           (* (* t_1 (fabs t)) (fabs t))))
         l)
        (/
         2.0
         (* (* (* (/ (fabs t) l) (* (fabs t) (/ t_1 l))) (tan k)) 2.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * fabs(t);
	double tmp;
	if (fabs(t) <= 1.66e-6) {
		tmp = ((cos(k) / ((fma(cos((k + k)), -0.5, 0.5) * k) * fabs(t))) * (l + l)) * (l / k);
	} else if (fabs(t) <= 8e+102) {
		tmp = (2.0 / (((tan(k) / l) * fma(k, (k / (fabs(t) * fabs(t))), 2.0)) * ((t_1 * fabs(t)) * fabs(t)))) * l;
	} else {
		tmp = 2.0 / ((((fabs(t) / l) * (fabs(t) * (t_1 / l))) * tan(k)) * 2.0);
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * abs(t))
	tmp = 0.0
	if (abs(t) <= 1.66e-6)
		tmp = Float64(Float64(Float64(cos(k) / Float64(Float64(fma(cos(Float64(k + k)), -0.5, 0.5) * k) * abs(t))) * Float64(l + l)) * Float64(l / k));
	elseif (abs(t) <= 8e+102)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(tan(k) / l) * fma(k, Float64(k / Float64(abs(t) * abs(t))), 2.0)) * Float64(Float64(t_1 * abs(t)) * abs(t)))) * l);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(abs(t) / l) * Float64(abs(t) * Float64(t_1 / l))) * tan(k)) * 2.0));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $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], 1.66e-6], N[(N[(N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * k), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 8e+102], N[(N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(k * N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.65999999999999999e-6

   1. Initial program 54.1%

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

   2. 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 59.7

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

   4. Applied rewrites 59.7%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 59.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 59.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 59.8

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

   8. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   10. Applied rewrites 69.1%

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

   if 1.65999999999999999e-6 < t < 7.99999999999999982e102

   1. Initial program 54.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.7

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

   3. Applied rewrites 66.7%

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

   5. Applied rewrites 56.3%

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

   if 7.99999999999999982e102 < t

   1. Initial program 54.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.7

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

   3. Applied rewrites 66.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.9

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

   5. Applied rewrites 75.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 75.9

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

   7. Applied rewrites 75.9%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 68.3%

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

Alternative 5: 87.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \left|k\right| \cdot t\\ \mathbf{if}\;\left|k\right| \leq 4.8 \cdot 10^{-162}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot t\right) \cdot t\_1} \cdot \ell\\ \mathbf{elif}\;\left|k\right| \leq 2.4 \cdot 10^{+22}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\sin \left(\left|k\right|\right) \cdot t}{\ell}\right)\right) \cdot \tan \left(\left|k\right|\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\cos \left(\left|k\right|\right)}{\left(\mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right) \cdot \left|k\right|\right) \cdot t} \cdot \left(\ell + \ell\right)\right) \cdot \frac{\ell}{\left|k\right|}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (fabs k) t)))
   (if (<= (fabs k) 4.8e-162)
     (* (/ l (* (* t_1 t) t_1)) l)
     (if (<= (fabs k) 2.4e+22)
       (/
        2.0
        (*
         (* (* (/ t l) (* t (/ (* (sin (fabs k)) t) l))) (tan (fabs k)))
         2.0))
       (*
        (*
         (/
          (cos (fabs k))
          (* (* (fma (cos (+ (fabs k) (fabs k))) -0.5 0.5) (fabs k)) t))
         (+ l l))
        (/ l (fabs k)))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(k) * t;
	double tmp;
	if (fabs(k) <= 4.8e-162) {
		tmp = (l / ((t_1 * t) * t_1)) * l;
	} else if (fabs(k) <= 2.4e+22) {
		tmp = 2.0 / ((((t / l) * (t * ((sin(fabs(k)) * t) / l))) * tan(fabs(k))) * 2.0);
	} else {
		tmp = ((cos(fabs(k)) / ((fma(cos((fabs(k) + fabs(k))), -0.5, 0.5) * fabs(k)) * t)) * (l + l)) * (l / fabs(k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(abs(k) * t)
	tmp = 0.0
	if (abs(k) <= 4.8e-162)
		tmp = Float64(Float64(l / Float64(Float64(t_1 * t) * t_1)) * l);
	elseif (abs(k) <= 2.4e+22)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t / l) * Float64(t * Float64(Float64(sin(abs(k)) * t) / l))) * tan(abs(k))) * 2.0));
	else
		tmp = Float64(Float64(Float64(cos(abs(k)) / Float64(Float64(fma(cos(Float64(abs(k) + abs(k))), -0.5, 0.5) * abs(k)) * t)) * Float64(l + l)) * Float64(l / abs(k)));
	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], 4.8e-162], N[(N[(l / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 2.4e+22], N[(2.0 / N[(N[(N[(N[(t / l), $MachinePrecision] * N[(t * N[(N[(N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision] * N[(l / N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.8000000000000004e-162

   1. Initial program 54.1%

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

   2. 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 50.3

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

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 59.7

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

      14. lift-pow.f64 N/A

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

      15. pow3 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 59.7

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

   6. Applied rewrites 59.7%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   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 59.7

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

      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. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 63.2

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

   8. Applied rewrites 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 66.4

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

   10. Applied rewrites 66.4%

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

   if 4.8000000000000004e-162 < k < 2.4e22

   1. Initial program 54.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.7

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

   3. Applied rewrites 66.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.9

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

   5. Applied rewrites 75.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 75.9

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

   7. Applied rewrites 75.9%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 68.3%

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

   if 2.4e22 < k

   1. Initial program 54.1%

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

   2. 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 59.7

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

   4. Applied rewrites 59.7%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 59.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 59.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 59.8

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

   8. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   10. Applied rewrites 69.1%

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

Alternative 6: 83.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \left|k\right| \cdot t\\ \mathbf{if}\;\left|k\right| \leq 4.8 \cdot 10^{-162}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot t\right) \cdot t\_1} \cdot \ell\\ \mathbf{elif}\;\left|k\right| \leq 2.4 \cdot 10^{+22}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\sin \left(\left|k\right|\right) \cdot t}{\ell}\right)\right) \cdot \tan \left(\left|k\right|\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell + \ell\right) \cdot \left(\ell \cdot \frac{\cos \left(\left|k\right|\right)}{\left(\left(\mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right) \cdot t\right) \cdot \left|k\right|\right) \cdot \left|k\right|}\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (fabs k) t)))
   (if (<= (fabs k) 4.8e-162)
     (* (/ l (* (* t_1 t) t_1)) l)
     (if (<= (fabs k) 2.4e+22)
       (/
        2.0
        (*
         (* (* (/ t l) (* t (/ (* (sin (fabs k)) t) l))) (tan (fabs k)))
         2.0))
       (*
        (+ l l)
        (*
         l
         (/
          (cos (fabs k))
          (*
           (* (* (fma (cos (+ (fabs k) (fabs k))) -0.5 0.5) t) (fabs k))
           (fabs k)))))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(k) * t;
	double tmp;
	if (fabs(k) <= 4.8e-162) {
		tmp = (l / ((t_1 * t) * t_1)) * l;
	} else if (fabs(k) <= 2.4e+22) {
		tmp = 2.0 / ((((t / l) * (t * ((sin(fabs(k)) * t) / l))) * tan(fabs(k))) * 2.0);
	} else {
		tmp = (l + l) * (l * (cos(fabs(k)) / (((fma(cos((fabs(k) + fabs(k))), -0.5, 0.5) * t) * fabs(k)) * fabs(k))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(abs(k) * t)
	tmp = 0.0
	if (abs(k) <= 4.8e-162)
		tmp = Float64(Float64(l / Float64(Float64(t_1 * t) * t_1)) * l);
	elseif (abs(k) <= 2.4e+22)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t / l) * Float64(t * Float64(Float64(sin(abs(k)) * t) / l))) * tan(abs(k))) * 2.0));
	else
		tmp = Float64(Float64(l + l) * Float64(l * Float64(cos(abs(k)) / Float64(Float64(Float64(fma(cos(Float64(abs(k) + abs(k))), -0.5, 0.5) * t) * abs(k)) * abs(k)))));
	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], 4.8e-162], N[(N[(l / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 2.4e+22], N[(2.0 / N[(N[(N[(N[(t / l), $MachinePrecision] * N[(t * N[(N[(N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(l + l), $MachinePrecision] * N[(l * N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.8000000000000004e-162

   1. Initial program 54.1%

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

   2. 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 50.3

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

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 59.7

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

      14. lift-pow.f64 N/A

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

      15. pow3 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 59.7

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

   6. Applied rewrites 59.7%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   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 59.7

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

      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. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 63.2

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

   8. Applied rewrites 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 66.4

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

   10. Applied rewrites 66.4%

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

   if 4.8000000000000004e-162 < k < 2.4e22

   1. Initial program 54.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.7

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

   3. Applied rewrites 66.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.9

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

   5. Applied rewrites 75.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 75.9

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

   7. Applied rewrites 75.9%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 68.3%

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

   if 2.4e22 < k

   1. Initial program 54.1%

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

   2. 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 59.7

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

   4. Applied rewrites 59.7%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 59.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 59.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. count-2-rev N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 65.5

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

   8. Applied rewrites 65.5%

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

Alternative 7: 77.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+22}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{k \cdot t}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\sin k \cdot t}{\ell}\right)\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 5e+22)
   (/
    2.0
    (* (* (* (/ t l) (* t (/ (* k t) l))) (tan k)) (fma (/ k t) (/ k t) 2.0)))
   (/ 2.0 (* (* (* (/ t l) (* t (/ (* (sin k) t) l))) (tan k)) 2.0))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e+22) {
		tmp = 2.0 / ((((t / l) * (t * ((k * t) / l))) * tan(k)) * fma((k / t), (k / t), 2.0));
	} else {
		tmp = 2.0 / ((((t / l) * (t * ((sin(k) * t) / l))) * tan(k)) * 2.0);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 5e+22)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t / l) * Float64(t * Float64(Float64(k * t) / l))) * tan(k)) * fma(Float64(k / t), Float64(k / t), 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t / l) * Float64(t * Float64(Float64(sin(k) * t) / l))) * tan(k)) * 2.0));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 5e+22], N[(2.0 / N[(N[(N[(N[(t / l), $MachinePrecision] * N[(t * N[(N[(k * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t / l), $MachinePrecision] * N[(t * N[(N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.9999999999999996e22

   1. Initial program 54.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.7

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

   3. Applied rewrites 66.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.9

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

   5. Applied rewrites 75.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 75.9

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

   7. Applied rewrites 75.9%

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

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 71.0

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

   10. Applied rewrites 71.0%

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

   if 4.9999999999999996e22 < (*.f64 l l)

   1. Initial program 54.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.7

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

   3. Applied rewrites 66.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.9

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

   5. Applied rewrites 75.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 75.9

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

   7. Applied rewrites 75.9%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 68.3%

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

Alternative 8: 70.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_1 := \left|k\right| \cdot t\\ \mathbf{if}\;\left|k\right| \leq 1.65 \cdot 10^{-105}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot t\right) \cdot t\_1} \cdot \ell\\ \mathbf{elif}\;\left|k\right| \leq 15000000000:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \left|k\right|\right) \cdot \tan \left(\left|k\right|\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos \left(\left|k\right|\right)}{\left(\left({\left(\left|k\right|\right)}^{2} \cdot t\right) \cdot \left|k\right|\right) \cdot \left|k\right|}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (fabs k) t)))
   (if (<= (fabs k) 1.65e-105)
     (* (/ l (* (* t_1 t) t_1)) l)
     (if (<= (fabs k) 15000000000.0)
       (/
        2.0
        (* (* (* (* (/ t l) (* (/ t l) t)) (fabs k)) (tan (fabs k))) 2.0))
       (*
        (* 2.0 (* l l))
        (/
         (cos (fabs k))
         (* (* (* (pow (fabs k) 2.0) t) (fabs k)) (fabs k))))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(k) * t;
	double tmp;
	if (fabs(k) <= 1.65e-105) {
		tmp = (l / ((t_1 * t) * t_1)) * l;
	} else if (fabs(k) <= 15000000000.0) {
		tmp = 2.0 / (((((t / l) * ((t / l) * t)) * fabs(k)) * tan(fabs(k))) * 2.0);
	} else {
		tmp = (2.0 * (l * l)) * (cos(fabs(k)) / (((pow(fabs(k), 2.0) * t) * fabs(k)) * fabs(k)));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(k) * t
    if (abs(k) <= 1.65d-105) then
        tmp = (l / ((t_1 * t) * t_1)) * l
    else if (abs(k) <= 15000000000.0d0) then
        tmp = 2.0d0 / (((((t / l) * ((t / l) * t)) * abs(k)) * tan(abs(k))) * 2.0d0)
    else
        tmp = (2.0d0 * (l * l)) * (cos(abs(k)) / ((((abs(k) ** 2.0d0) * t) * abs(k)) * abs(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.abs(k) * t;
	double tmp;
	if (Math.abs(k) <= 1.65e-105) {
		tmp = (l / ((t_1 * t) * t_1)) * l;
	} else if (Math.abs(k) <= 15000000000.0) {
		tmp = 2.0 / (((((t / l) * ((t / l) * t)) * Math.abs(k)) * Math.tan(Math.abs(k))) * 2.0);
	} else {
		tmp = (2.0 * (l * l)) * (Math.cos(Math.abs(k)) / (((Math.pow(Math.abs(k), 2.0) * t) * Math.abs(k)) * Math.abs(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(k) * t
	tmp = 0
	if math.fabs(k) <= 1.65e-105:
		tmp = (l / ((t_1 * t) * t_1)) * l
	elif math.fabs(k) <= 15000000000.0:
		tmp = 2.0 / (((((t / l) * ((t / l) * t)) * math.fabs(k)) * math.tan(math.fabs(k))) * 2.0)
	else:
		tmp = (2.0 * (l * l)) * (math.cos(math.fabs(k)) / (((math.pow(math.fabs(k), 2.0) * t) * math.fabs(k)) * math.fabs(k)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(k) * t)
	tmp = 0.0
	if (abs(k) <= 1.65e-105)
		tmp = Float64(Float64(l / Float64(Float64(t_1 * t) * t_1)) * l);
	elseif (abs(k) <= 15000000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t / l) * Float64(Float64(t / l) * t)) * abs(k)) * tan(abs(k))) * 2.0));
	else
		tmp = Float64(Float64(2.0 * Float64(l * l)) * Float64(cos(abs(k)) / Float64(Float64(Float64((abs(k) ^ 2.0) * t) * abs(k)) * abs(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(k) * t;
	tmp = 0.0;
	if (abs(k) <= 1.65e-105)
		tmp = (l / ((t_1 * t) * t_1)) * l;
	elseif (abs(k) <= 15000000000.0)
		tmp = 2.0 / (((((t / l) * ((t / l) * t)) * abs(k)) * tan(abs(k))) * 2.0);
	else
		tmp = (2.0 * (l * l)) * (cos(abs(k)) / ((((abs(k) ^ 2.0) * t) * abs(k)) * abs(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 1.65e-105], N[(N[(l / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 15000000000.0], N[(2.0 / N[(N[(N[(N[(N[(t / l), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.6499999999999999e-105

   1. Initial program 54.1%

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

   2. 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 50.3

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

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 59.7

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

      14. lift-pow.f64 N/A

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

      15. pow3 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 59.7

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

   6. Applied rewrites 59.7%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   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 59.7

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

      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. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 63.2

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

   8. Applied rewrites 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 66.4

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

   10. Applied rewrites 66.4%

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

   if 1.6499999999999999e-105 < k < 1.5e10

   1. Initial program 54.1%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 54.2%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 52.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. cube-mult N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 62.7

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

   9. Applied rewrites 62.7%

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

   if 1.5e10 < k

   1. Initial program 54.1%

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

   2. 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 59.7

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

   4. Applied rewrites 59.7%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 59.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 59.0%

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

   7. Taylor expanded in k around 0

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

      1. lower-pow.f64 53.7

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

   9. Applied rewrites 53.7%

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

Alternative 9: 70.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_1 := \left|k\right| \cdot t\\ \mathbf{if}\;\left|k\right| \leq 1.65 \cdot 10^{-105}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot t\right) \cdot t\_1} \cdot \ell\\ \mathbf{elif}\;\left|k\right| \leq 15000000000:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \left|k\right|\right) \cdot \tan \left(\left|k\right|\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos \left(\left|k\right|\right)}{\left({\left(\left|k\right|\right)}^{3} \cdot t\right) \cdot \left|k\right|}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (fabs k) t)))
   (if (<= (fabs k) 1.65e-105)
     (* (/ l (* (* t_1 t) t_1)) l)
     (if (<= (fabs k) 15000000000.0)
       (/
        2.0
        (* (* (* (* (/ t l) (* (/ t l) t)) (fabs k)) (tan (fabs k))) 2.0))
       (*
        (* 2.0 (* l l))
        (/ (cos (fabs k)) (* (* (pow (fabs k) 3.0) t) (fabs k))))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(k) * t;
	double tmp;
	if (fabs(k) <= 1.65e-105) {
		tmp = (l / ((t_1 * t) * t_1)) * l;
	} else if (fabs(k) <= 15000000000.0) {
		tmp = 2.0 / (((((t / l) * ((t / l) * t)) * fabs(k)) * tan(fabs(k))) * 2.0);
	} else {
		tmp = (2.0 * (l * l)) * (cos(fabs(k)) / ((pow(fabs(k), 3.0) * t) * fabs(k)));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(k) * t
    if (abs(k) <= 1.65d-105) then
        tmp = (l / ((t_1 * t) * t_1)) * l
    else if (abs(k) <= 15000000000.0d0) then
        tmp = 2.0d0 / (((((t / l) * ((t / l) * t)) * abs(k)) * tan(abs(k))) * 2.0d0)
    else
        tmp = (2.0d0 * (l * l)) * (cos(abs(k)) / (((abs(k) ** 3.0d0) * t) * abs(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.abs(k) * t;
	double tmp;
	if (Math.abs(k) <= 1.65e-105) {
		tmp = (l / ((t_1 * t) * t_1)) * l;
	} else if (Math.abs(k) <= 15000000000.0) {
		tmp = 2.0 / (((((t / l) * ((t / l) * t)) * Math.abs(k)) * Math.tan(Math.abs(k))) * 2.0);
	} else {
		tmp = (2.0 * (l * l)) * (Math.cos(Math.abs(k)) / ((Math.pow(Math.abs(k), 3.0) * t) * Math.abs(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(k) * t
	tmp = 0
	if math.fabs(k) <= 1.65e-105:
		tmp = (l / ((t_1 * t) * t_1)) * l
	elif math.fabs(k) <= 15000000000.0:
		tmp = 2.0 / (((((t / l) * ((t / l) * t)) * math.fabs(k)) * math.tan(math.fabs(k))) * 2.0)
	else:
		tmp = (2.0 * (l * l)) * (math.cos(math.fabs(k)) / ((math.pow(math.fabs(k), 3.0) * t) * math.fabs(k)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(k) * t)
	tmp = 0.0
	if (abs(k) <= 1.65e-105)
		tmp = Float64(Float64(l / Float64(Float64(t_1 * t) * t_1)) * l);
	elseif (abs(k) <= 15000000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t / l) * Float64(Float64(t / l) * t)) * abs(k)) * tan(abs(k))) * 2.0));
	else
		tmp = Float64(Float64(2.0 * Float64(l * l)) * Float64(cos(abs(k)) / Float64(Float64((abs(k) ^ 3.0) * t) * abs(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(k) * t;
	tmp = 0.0;
	if (abs(k) <= 1.65e-105)
		tmp = (l / ((t_1 * t) * t_1)) * l;
	elseif (abs(k) <= 15000000000.0)
		tmp = 2.0 / (((((t / l) * ((t / l) * t)) * abs(k)) * tan(abs(k))) * 2.0);
	else
		tmp = (2.0 * (l * l)) * (cos(abs(k)) / (((abs(k) ^ 3.0) * t) * abs(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 1.65e-105], N[(N[(l / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 15000000000.0], N[(2.0 / N[(N[(N[(N[(N[(t / l), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] / N[(N[(N[Power[N[Abs[k], $MachinePrecision], 3.0], $MachinePrecision] * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.6499999999999999e-105

   1. Initial program 54.1%

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

   2. 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 50.3

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

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 59.7

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

      14. lift-pow.f64 N/A

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

      15. pow3 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 59.7

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

   6. Applied rewrites 59.7%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   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 59.7

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

      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. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 63.2

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

   8. Applied rewrites 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 66.4

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

   10. Applied rewrites 66.4%

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

   if 1.6499999999999999e-105 < k < 1.5e10

   1. Initial program 54.1%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 54.2%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 52.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. cube-mult N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 62.7

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

   9. Applied rewrites 62.7%

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

   if 1.5e10 < k

   1. Initial program 54.1%

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

   2. 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 59.7

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

   4. Applied rewrites 59.7%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 59.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 59.0%

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

   7. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 53.2

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

   9. Applied rewrites 53.2%

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

Alternative 10: 70.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_1 := \left|k\right| \cdot t\\ \mathbf{if}\;\left|k\right| \leq 1.65 \cdot 10^{-105}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot t\right) \cdot t\_1} \cdot \ell\\ \mathbf{elif}\;\left|k\right| \leq 15000000000:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \left|k\right|\right) \cdot \tan \left(\left|k\right|\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos \left(\left|k\right|\right)}{{\left(\left|k\right|\right)}^{4} \cdot t}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (fabs k) t)))
   (if (<= (fabs k) 1.65e-105)
     (* (/ l (* (* t_1 t) t_1)) l)
     (if (<= (fabs k) 15000000000.0)
       (/
        2.0
        (* (* (* (* (/ t l) (* (/ t l) t)) (fabs k)) (tan (fabs k))) 2.0))
       (* (* 2.0 (* l l)) (/ (cos (fabs k)) (* (pow (fabs k) 4.0) t)))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(k) * t;
	double tmp;
	if (fabs(k) <= 1.65e-105) {
		tmp = (l / ((t_1 * t) * t_1)) * l;
	} else if (fabs(k) <= 15000000000.0) {
		tmp = 2.0 / (((((t / l) * ((t / l) * t)) * fabs(k)) * tan(fabs(k))) * 2.0);
	} else {
		tmp = (2.0 * (l * l)) * (cos(fabs(k)) / (pow(fabs(k), 4.0) * t));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(k) * t
    if (abs(k) <= 1.65d-105) then
        tmp = (l / ((t_1 * t) * t_1)) * l
    else if (abs(k) <= 15000000000.0d0) then
        tmp = 2.0d0 / (((((t / l) * ((t / l) * t)) * abs(k)) * tan(abs(k))) * 2.0d0)
    else
        tmp = (2.0d0 * (l * l)) * (cos(abs(k)) / ((abs(k) ** 4.0d0) * t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.abs(k) * t;
	double tmp;
	if (Math.abs(k) <= 1.65e-105) {
		tmp = (l / ((t_1 * t) * t_1)) * l;
	} else if (Math.abs(k) <= 15000000000.0) {
		tmp = 2.0 / (((((t / l) * ((t / l) * t)) * Math.abs(k)) * Math.tan(Math.abs(k))) * 2.0);
	} else {
		tmp = (2.0 * (l * l)) * (Math.cos(Math.abs(k)) / (Math.pow(Math.abs(k), 4.0) * t));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(k) * t
	tmp = 0
	if math.fabs(k) <= 1.65e-105:
		tmp = (l / ((t_1 * t) * t_1)) * l
	elif math.fabs(k) <= 15000000000.0:
		tmp = 2.0 / (((((t / l) * ((t / l) * t)) * math.fabs(k)) * math.tan(math.fabs(k))) * 2.0)
	else:
		tmp = (2.0 * (l * l)) * (math.cos(math.fabs(k)) / (math.pow(math.fabs(k), 4.0) * t))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(k) * t)
	tmp = 0.0
	if (abs(k) <= 1.65e-105)
		tmp = Float64(Float64(l / Float64(Float64(t_1 * t) * t_1)) * l);
	elseif (abs(k) <= 15000000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t / l) * Float64(Float64(t / l) * t)) * abs(k)) * tan(abs(k))) * 2.0));
	else
		tmp = Float64(Float64(2.0 * Float64(l * l)) * Float64(cos(abs(k)) / Float64((abs(k) ^ 4.0) * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(k) * t;
	tmp = 0.0;
	if (abs(k) <= 1.65e-105)
		tmp = (l / ((t_1 * t) * t_1)) * l;
	elseif (abs(k) <= 15000000000.0)
		tmp = 2.0 / (((((t / l) * ((t / l) * t)) * abs(k)) * tan(abs(k))) * 2.0);
	else
		tmp = (2.0 * (l * l)) * (cos(abs(k)) / ((abs(k) ^ 4.0) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 1.65e-105], N[(N[(l / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 15000000000.0], N[(2.0 / N[(N[(N[(N[(N[(t / l), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] / N[(N[Power[N[Abs[k], $MachinePrecision], 4.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.6499999999999999e-105

   1. Initial program 54.1%

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

   2. 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 50.3

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

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 59.7

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

      14. lift-pow.f64 N/A

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

      15. pow3 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 59.7

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

   6. Applied rewrites 59.7%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   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 59.7

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

      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. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      13. lower-*.f64 63.2

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 63.2%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lower-*.f64 66.4

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 66.4%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   if 1.6499999999999999e-105 < k < 1.5e10

   1. Initial program 54.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{2}} \]
   3. Step-by-step derivation

   4. Applied rewrites 54.2%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{2}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{k}\right) \cdot \tan k\right) \cdot 2} \]
   6. Step-by-step derivation

   7. Applied rewrites 52.8%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{k}\right) \cdot \tan k\right) \cdot 2} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      4. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot \color{blue}{\left(t \cdot t\right)}}{\ell \cdot \ell} \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      6. times-frac N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)} \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t \cdot t}{\ell}\right) \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)} \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot \frac{\color{blue}{t \cdot t}}{\ell}\right) \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}\right) \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot t\right)\right) \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      12. lower-*.f64 62.7

          \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}\right) \cdot k\right) \cdot \tan k\right) \cdot 2} \]

   9. Applied rewrites 62.7%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot t\right)\right)} \cdot k\right) \cdot \tan k\right) \cdot 2} \]

   if 1.5e10 < k

   1. Initial program 54.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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 59.7

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 59.7%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. associate-/l* N/A

         \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \color{blue}{\frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(2 \cdot {\ell}^{2}\right) \cdot \color{blue}{\frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot {\ell}^{2}\right) \cdot \color{blue}{\frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot {\ell}^{2}\right) \cdot \frac{\color{blue}{\cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. pow2 N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      11. lower-/.f64 59.6

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      13. *-commutative N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

      14. lift-pow.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{\color{blue}{2}}} \]

      15. unpow2 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(t \cdot {\sin k}^{2}\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      16. associate-*r* N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot \color{blue}{k}} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot \color{blue}{k}} \]

   6. Applied rewrites 59.0%

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\cos k}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{{k}^{4} \cdot \color{blue}{t}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{{k}^{4} \cdot t} \]

      2. lower-pow.f64 52.6

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{{k}^{4} \cdot t} \]

   9. Applied rewrites 52.6%

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{{k}^{4} \cdot \color{blue}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 70.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_1 := \left|k\right| \cdot t\\ \mathbf{if}\;\left|k\right| \leq 1.65 \cdot 10^{-105}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot t\right) \cdot t\_1} \cdot \ell\\ \mathbf{elif}\;\left|k\right| \leq 3.4 \cdot 10^{+28}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \left|k\right|\right) \cdot \tan \left(\left|k\right|\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell + \ell\right) \cdot \ell}{\left|k\right|} \cdot \frac{1}{\left(\mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right) \cdot t\right) \cdot \left|k\right|}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (fabs k) t)))
   (if (<= (fabs k) 1.65e-105)
     (* (/ l (* (* t_1 t) t_1)) l)
     (if (<= (fabs k) 3.4e+28)
       (/
        2.0
        (* (* (* (* (/ t l) (* (/ t l) t)) (fabs k)) (tan (fabs k))) 2.0))
       (*
        (/ (* (+ l l) l) (fabs k))
        (/
         1.0
         (* (* (fma (cos (+ (fabs k) (fabs k))) -0.5 0.5) t) (fabs k))))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(k) * t;
	double tmp;
	if (fabs(k) <= 1.65e-105) {
		tmp = (l / ((t_1 * t) * t_1)) * l;
	} else if (fabs(k) <= 3.4e+28) {
		tmp = 2.0 / (((((t / l) * ((t / l) * t)) * fabs(k)) * tan(fabs(k))) * 2.0);
	} else {
		tmp = (((l + l) * l) / fabs(k)) * (1.0 / ((fma(cos((fabs(k) + fabs(k))), -0.5, 0.5) * t) * fabs(k)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(abs(k) * t)
	tmp = 0.0
	if (abs(k) <= 1.65e-105)
		tmp = Float64(Float64(l / Float64(Float64(t_1 * t) * t_1)) * l);
	elseif (abs(k) <= 3.4e+28)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t / l) * Float64(Float64(t / l) * t)) * abs(k)) * tan(abs(k))) * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(l + l) * l) / abs(k)) * Float64(1.0 / Float64(Float64(fma(cos(Float64(abs(k) + abs(k))), -0.5, 0.5) * t) * abs(k))));
	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], 1.65e-105], N[(N[(l / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 3.4e+28], N[(2.0 / N[(N[(N[(N[(N[(t / l), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l + l), $MachinePrecision] * l), $MachinePrecision] / N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.6499999999999999e-105

   1. Initial program 54.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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 50.3

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.1

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.7

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lift-*.f64 59.7

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.7%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   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 59.7

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      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. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      13. lower-*.f64 63.2

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 63.2%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lower-*.f64 66.4

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 66.4%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   if 1.6499999999999999e-105 < k < 3.4e28

   1. Initial program 54.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{2}} \]
   3. Step-by-step derivation

   4. Applied rewrites 54.2%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{2}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{k}\right) \cdot \tan k\right) \cdot 2} \]
   6. Step-by-step derivation

   7. Applied rewrites 52.8%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{k}\right) \cdot \tan k\right) \cdot 2} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      4. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot \color{blue}{\left(t \cdot t\right)}}{\ell \cdot \ell} \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      6. times-frac N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)} \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t \cdot t}{\ell}\right) \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)} \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot \frac{\color{blue}{t \cdot t}}{\ell}\right) \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}\right) \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot t\right)\right) \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      12. lower-*.f64 62.7

          \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}\right) \cdot k\right) \cdot \tan k\right) \cdot 2} \]

   9. Applied rewrites 62.7%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot t\right)\right)} \cdot k\right) \cdot \tan k\right) \cdot 2} \]

   if 3.4e28 < k

   1. Initial program 54.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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 59.7

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 59.7%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. associate-/l* N/A

         \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \color{blue}{\frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(2 \cdot {\ell}^{2}\right) \cdot \color{blue}{\frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot {\ell}^{2}\right) \cdot \color{blue}{\frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot {\ell}^{2}\right) \cdot \frac{\color{blue}{\cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. pow2 N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      11. lower-/.f64 59.6

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      13. *-commutative N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

      14. lift-pow.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{\color{blue}{2}}} \]

      15. unpow2 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(t \cdot {\sin k}^{2}\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      16. associate-*r* N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot \color{blue}{k}} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot \color{blue}{k}} \]

   6. Applied rewrites 59.0%

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\cos k}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\cos k}{\left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\color{blue}{\left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k \cdot \color{blue}{\left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right)}} \]

      6. times-frac N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k} \cdot \color{blue}{\frac{\cos k}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k} \cdot \color{blue}{\frac{\cos k}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k} \cdot \frac{\color{blue}{\cos k}}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k} \cdot \frac{\cos \color{blue}{k}}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k} \cdot \frac{\cos k}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \]

      11. associate-*r* N/A

          \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{k} \cdot \frac{\cos \color{blue}{k}}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{k} \cdot \frac{\cos \color{blue}{k}}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \]

      13. count-2-rev N/A

          \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{k} \cdot \frac{\cos k}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \]

      14. lower-+.f64 N/A

          \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{k} \cdot \frac{\cos k}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \]

      15. lower-/.f64 59.8

          \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{k} \cdot \frac{\cos k}{\color{blue}{\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k}} \]

   8. Applied rewrites 59.8%

      \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{k} \cdot \color{blue}{\frac{\cos k}{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right) \cdot k}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{k} \cdot \frac{1}{\color{blue}{\left(\mathsf{fma}\left(\cos \left(k + k\right), \frac{-1}{2}, \frac{1}{2}\right) \cdot t\right)} \cdot k} \]
   10. Step-by-step derivation

   11. Applied rewrites 50.2%

       \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{k} \cdot \frac{1}{\color{blue}{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right)} \cdot k} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 70.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_1 := \left|k\right| \cdot t\\ \mathbf{if}\;\left|k\right| \leq 1.65 \cdot 10^{-105}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot t\right) \cdot t\_1} \cdot \ell\\ \mathbf{elif}\;\left|k\right| \leq 3.4 \cdot 10^{+28}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \left|k\right|\right) \cdot \tan \left(\left|k\right|\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell + \ell\right) \cdot \ell}{\left|k\right|} \cdot \frac{1}{{\left(\left|k\right|\right)}^{3} \cdot t}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (fabs k) t)))
   (if (<= (fabs k) 1.65e-105)
     (* (/ l (* (* t_1 t) t_1)) l)
     (if (<= (fabs k) 3.4e+28)
       (/
        2.0
        (* (* (* (* (/ t l) (* (/ t l) t)) (fabs k)) (tan (fabs k))) 2.0))
       (* (/ (* (+ l l) l) (fabs k)) (/ 1.0 (* (pow (fabs k) 3.0) t)))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(k) * t;
	double tmp;
	if (fabs(k) <= 1.65e-105) {
		tmp = (l / ((t_1 * t) * t_1)) * l;
	} else if (fabs(k) <= 3.4e+28) {
		tmp = 2.0 / (((((t / l) * ((t / l) * t)) * fabs(k)) * tan(fabs(k))) * 2.0);
	} else {
		tmp = (((l + l) * l) / fabs(k)) * (1.0 / (pow(fabs(k), 3.0) * t));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(k) * t
    if (abs(k) <= 1.65d-105) then
        tmp = (l / ((t_1 * t) * t_1)) * l
    else if (abs(k) <= 3.4d+28) then
        tmp = 2.0d0 / (((((t / l) * ((t / l) * t)) * abs(k)) * tan(abs(k))) * 2.0d0)
    else
        tmp = (((l + l) * l) / abs(k)) * (1.0d0 / ((abs(k) ** 3.0d0) * t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.abs(k) * t;
	double tmp;
	if (Math.abs(k) <= 1.65e-105) {
		tmp = (l / ((t_1 * t) * t_1)) * l;
	} else if (Math.abs(k) <= 3.4e+28) {
		tmp = 2.0 / (((((t / l) * ((t / l) * t)) * Math.abs(k)) * Math.tan(Math.abs(k))) * 2.0);
	} else {
		tmp = (((l + l) * l) / Math.abs(k)) * (1.0 / (Math.pow(Math.abs(k), 3.0) * t));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(k) * t
	tmp = 0
	if math.fabs(k) <= 1.65e-105:
		tmp = (l / ((t_1 * t) * t_1)) * l
	elif math.fabs(k) <= 3.4e+28:
		tmp = 2.0 / (((((t / l) * ((t / l) * t)) * math.fabs(k)) * math.tan(math.fabs(k))) * 2.0)
	else:
		tmp = (((l + l) * l) / math.fabs(k)) * (1.0 / (math.pow(math.fabs(k), 3.0) * t))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(k) * t)
	tmp = 0.0
	if (abs(k) <= 1.65e-105)
		tmp = Float64(Float64(l / Float64(Float64(t_1 * t) * t_1)) * l);
	elseif (abs(k) <= 3.4e+28)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t / l) * Float64(Float64(t / l) * t)) * abs(k)) * tan(abs(k))) * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(l + l) * l) / abs(k)) * Float64(1.0 / Float64((abs(k) ^ 3.0) * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(k) * t;
	tmp = 0.0;
	if (abs(k) <= 1.65e-105)
		tmp = (l / ((t_1 * t) * t_1)) * l;
	elseif (abs(k) <= 3.4e+28)
		tmp = 2.0 / (((((t / l) * ((t / l) * t)) * abs(k)) * tan(abs(k))) * 2.0);
	else
		tmp = (((l + l) * l) / abs(k)) * (1.0 / ((abs(k) ^ 3.0) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 1.65e-105], N[(N[(l / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 3.4e+28], N[(2.0 / N[(N[(N[(N[(N[(t / l), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l + l), $MachinePrecision] * l), $MachinePrecision] / N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[Power[N[Abs[k], $MachinePrecision], 3.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.6499999999999999e-105

   1. Initial program 54.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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 50.3

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.1

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.7

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lift-*.f64 59.7

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.7%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   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 59.7

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      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. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      13. lower-*.f64 63.2

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 63.2%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lower-*.f64 66.4

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 66.4%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   if 1.6499999999999999e-105 < k < 3.4e28

   1. Initial program 54.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{2}} \]
   3. Step-by-step derivation

   4. Applied rewrites 54.2%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{2}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{k}\right) \cdot \tan k\right) \cdot 2} \]
   6. Step-by-step derivation

   7. Applied rewrites 52.8%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{k}\right) \cdot \tan k\right) \cdot 2} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      4. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot \color{blue}{\left(t \cdot t\right)}}{\ell \cdot \ell} \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      6. times-frac N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)} \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t \cdot t}{\ell}\right) \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)} \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot \frac{\color{blue}{t \cdot t}}{\ell}\right) \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}\right) \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot t\right)\right) \cdot k\right) \cdot \tan k\right) \cdot 2} \]

      12. lower-*.f64 62.7

          \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}\right) \cdot k\right) \cdot \tan k\right) \cdot 2} \]

   9. Applied rewrites 62.7%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot t\right)\right)} \cdot k\right) \cdot \tan k\right) \cdot 2} \]

   if 3.4e28 < k

   1. Initial program 54.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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 59.7

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 59.7%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. associate-/l* N/A

         \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \color{blue}{\frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(2 \cdot {\ell}^{2}\right) \cdot \color{blue}{\frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot {\ell}^{2}\right) \cdot \color{blue}{\frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot {\ell}^{2}\right) \cdot \frac{\color{blue}{\cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. pow2 N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      11. lower-/.f64 59.6

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      13. *-commutative N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

      14. lift-pow.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{\color{blue}{2}}} \]

      15. unpow2 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(t \cdot {\sin k}^{2}\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      16. associate-*r* N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot \color{blue}{k}} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot \color{blue}{k}} \]

   6. Applied rewrites 59.0%

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\cos k}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\cos k}{\left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\color{blue}{\left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k \cdot \color{blue}{\left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right)}} \]

      6. times-frac N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k} \cdot \color{blue}{\frac{\cos k}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k} \cdot \color{blue}{\frac{\cos k}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k} \cdot \frac{\color{blue}{\cos k}}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k} \cdot \frac{\cos \color{blue}{k}}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k} \cdot \frac{\cos k}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \]

      11. associate-*r* N/A

          \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{k} \cdot \frac{\cos \color{blue}{k}}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{k} \cdot \frac{\cos \color{blue}{k}}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \]

      13. count-2-rev N/A

          \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{k} \cdot \frac{\cos k}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \]

      14. lower-+.f64 N/A

          \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{k} \cdot \frac{\cos k}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \]

      15. lower-/.f64 59.8

          \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{k} \cdot \frac{\cos k}{\color{blue}{\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k}} \]

   8. Applied rewrites 59.8%

      \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{k} \cdot \color{blue}{\frac{\cos k}{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right) \cdot k}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{k} \cdot \frac{1}{\color{blue}{{k}^{3} \cdot t}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{k} \cdot \frac{1}{{k}^{3} \cdot \color{blue}{t}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{k} \cdot \frac{1}{{k}^{3} \cdot t} \]

       3. lower-pow.f64 52.2

          \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{k} \cdot \frac{1}{{k}^{3} \cdot t} \]

   11. Applied rewrites 52.2%

       \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{k} \cdot \frac{1}{\color{blue}{{k}^{3} \cdot t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 69.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|k\right|}{t}\\ t_2 := \left|k\right| \cdot t\\ \mathbf{if}\;\left|k\right| \leq 3.8 \cdot 10^{-105}:\\ \;\;\;\;\frac{\ell}{\left(t\_2 \cdot t\right) \cdot t\_2} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t\_2}{\ell}\right)\right) \cdot \tan \left(\left|k\right|\right)\right) \cdot \mathsf{fma}\left(t\_1, t\_1, 2\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (fabs k) t)) (t_2 (* (fabs k) t)))
   (if (<= (fabs k) 3.8e-105)
     (* (/ l (* (* t_2 t) t_2)) l)
     (/
      2.0
      (* (* (* (/ t l) (* t (/ t_2 l))) (tan (fabs k))) (fma t_1 t_1 2.0))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(k) / t;
	double t_2 = fabs(k) * t;
	double tmp;
	if (fabs(k) <= 3.8e-105) {
		tmp = (l / ((t_2 * t) * t_2)) * l;
	} else {
		tmp = 2.0 / ((((t / l) * (t * (t_2 / l))) * tan(fabs(k))) * fma(t_1, t_1, 2.0));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(abs(k) / t)
	t_2 = Float64(abs(k) * t)
	tmp = 0.0
	if (abs(k) <= 3.8e-105)
		tmp = Float64(Float64(l / Float64(Float64(t_2 * t) * t_2)) * l);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t / l) * Float64(t * Float64(t_2 / l))) * tan(abs(k))) * fma(t_1, t_1, 2.0)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[k], $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 3.8e-105], N[(N[(l / N[(N[(t$95$2 * t), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t / l), $MachinePrecision] * N[(t * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.7999999999999998e-105

   1. Initial program 54.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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 50.3

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.1

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.7

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lift-*.f64 59.7

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.7%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   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 59.7

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      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. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      13. lower-*.f64 63.2

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 63.2%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lower-*.f64 66.4

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 66.4%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   if 3.7999999999999998e-105 < k

   1. Initial program 54.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3} \cdot \sin k}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. cube-mult N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t \cdot \left(\left(t \cdot t\right) \cdot \sin k\right)}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t}{\ell}} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{\color{blue}{\left(t \cdot t\right) \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 66.7

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{\color{blue}{\left(t \cdot t\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 66.7%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{\color{blue}{\left(t \cdot t\right) \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{\color{blue}{\left(t \cdot t\right)} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{\color{blue}{t \cdot \left(t \cdot \sin k\right)}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \frac{t \cdot \sin k}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \frac{t \cdot \sin k}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 75.9

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 75.9%

      \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \frac{\sin k \cdot t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\sin k \cdot t}{\ell}\right)\right) \cdot \tan k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\sin k \cdot t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\sin k \cdot t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\sin k \cdot t}{\ell}\right)\right) \cdot \tan k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)\right)}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\sin k \cdot t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + \left(1 + 1\right)\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\sin k \cdot t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + \left(1 + 1\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\sin k \cdot t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + \color{blue}{2}\right)} \]

      8. lower-fma.f64 75.9

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\sin k \cdot t}{\ell}\right)\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   7. Applied rewrites 75.9%

      \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\sin k \cdot t}{\ell}\right)\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \color{blue}{\frac{k \cdot t}{\ell}}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{k \cdot t}{\color{blue}{\ell}}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

      2. lower-*.f64 71.0

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{k \cdot t}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

   10. Applied rewrites 71.0%

       \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \color{blue}{\frac{k \cdot t}{\ell}}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 68.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} t_1 := k \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 7 \cdot 10^{-50}:\\ \;\;\;\;\frac{\left(\ell + \ell\right) \cdot \ell}{k} \cdot \frac{1}{{k}^{3} \cdot \left|t\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot \left|t\right|\right) \cdot t\_1} \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 7e-50)
      (* (/ (* (+ l l) l) k) (/ 1.0 (* (pow k 3.0) (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) <= 7e-50) {
		tmp = (((l + l) * l) / k) * (1.0 / (pow(k, 3.0) * 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) <= 7e-50) {
		tmp = (((l + l) * l) / k) * (1.0 / (Math.pow(k, 3.0) * 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) <= 7e-50:
		tmp = (((l + l) * l) / k) * (1.0 / (math.pow(k, 3.0) * 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) <= 7e-50)
		tmp = Float64(Float64(Float64(Float64(l + l) * l) / k) * Float64(1.0 / Float64((k ^ 3.0) * 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) <= 7e-50)
		tmp = (((l + l) * l) / k) * (1.0 / ((k ^ 3.0) * 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], 7e-50], N[(N[(N[(N[(l + l), $MachinePrecision] * l), $MachinePrecision] / k), $MachinePrecision] * N[(1.0 / N[(N[Power[k, 3.0], $MachinePrecision] * N[Abs[t], $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 2 regimes

2. if t < 6.99999999999999993e-50

   1. Initial program 54.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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 59.7

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 59.7%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. associate-/l* N/A

         \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \color{blue}{\frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(2 \cdot {\ell}^{2}\right) \cdot \color{blue}{\frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot {\ell}^{2}\right) \cdot \color{blue}{\frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot {\ell}^{2}\right) \cdot \frac{\color{blue}{\cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. pow2 N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      11. lower-/.f64 59.6

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      13. *-commutative N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

      14. lift-pow.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{\color{blue}{2}}} \]

      15. unpow2 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(t \cdot {\sin k}^{2}\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      16. associate-*r* N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot \color{blue}{k}} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot \color{blue}{k}} \]

   6. Applied rewrites 59.0%

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\cos k}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\cos k}{\left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\color{blue}{\left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{k \cdot \color{blue}{\left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right)}} \]

      6. times-frac N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k} \cdot \color{blue}{\frac{\cos k}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k} \cdot \color{blue}{\frac{\cos k}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k} \cdot \frac{\color{blue}{\cos k}}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k} \cdot \frac{\cos \color{blue}{k}}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{k} \cdot \frac{\cos k}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \]

      11. associate-*r* N/A

          \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{k} \cdot \frac{\cos \color{blue}{k}}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{k} \cdot \frac{\cos \color{blue}{k}}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \]

      13. count-2-rev N/A

          \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{k} \cdot \frac{\cos k}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \]

      14. lower-+.f64 N/A

          \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{k} \cdot \frac{\cos k}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \]

      15. lower-/.f64 59.8

          \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{k} \cdot \frac{\cos k}{\color{blue}{\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k}} \]

   8. Applied rewrites 59.8%

      \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{k} \cdot \color{blue}{\frac{\cos k}{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right) \cdot k}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{k} \cdot \frac{1}{\color{blue}{{k}^{3} \cdot t}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{k} \cdot \frac{1}{{k}^{3} \cdot \color{blue}{t}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{k} \cdot \frac{1}{{k}^{3} \cdot t} \]

       3. lower-pow.f64 52.2

          \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{k} \cdot \frac{1}{{k}^{3} \cdot t} \]

   11. Applied rewrites 52.2%

       \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{k} \cdot \frac{1}{\color{blue}{{k}^{3} \cdot t}} \]

   if 6.99999999999999993e-50 < t

   1. Initial program 54.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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 50.3

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.1

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.7

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lift-*.f64 59.7

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.7%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   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 59.7

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      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. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      13. lower-*.f64 63.2

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 63.2%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lower-*.f64 66.4

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 66.4%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 68.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} t_1 := \left|k\right| \cdot t\\ \mathbf{if}\;\left|k\right| \leq 2.4 \cdot 10^{+22}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot t\right) \cdot t\_1} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{{\left(\left|k\right|\right)}^{4} \cdot t}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (fabs k) t)))
   (if (<= (fabs k) 2.4e+22)
     (* (/ l (* (* t_1 t) t_1)) l)
     (* (* 2.0 (* l l)) (/ 1.0 (* (pow (fabs k) 4.0) t))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(k) * t;
	double tmp;
	if (fabs(k) <= 2.4e+22) {
		tmp = (l / ((t_1 * t) * t_1)) * l;
	} else {
		tmp = (2.0 * (l * l)) * (1.0 / (pow(fabs(k), 4.0) * t));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(k) * t
    if (abs(k) <= 2.4d+22) then
        tmp = (l / ((t_1 * t) * t_1)) * l
    else
        tmp = (2.0d0 * (l * l)) * (1.0d0 / ((abs(k) ** 4.0d0) * t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.abs(k) * t;
	double tmp;
	if (Math.abs(k) <= 2.4e+22) {
		tmp = (l / ((t_1 * t) * t_1)) * l;
	} else {
		tmp = (2.0 * (l * l)) * (1.0 / (Math.pow(Math.abs(k), 4.0) * t));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(k) * t
	tmp = 0
	if math.fabs(k) <= 2.4e+22:
		tmp = (l / ((t_1 * t) * t_1)) * l
	else:
		tmp = (2.0 * (l * l)) * (1.0 / (math.pow(math.fabs(k), 4.0) * t))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(k) * t)
	tmp = 0.0
	if (abs(k) <= 2.4e+22)
		tmp = Float64(Float64(l / Float64(Float64(t_1 * t) * t_1)) * l);
	else
		tmp = Float64(Float64(2.0 * Float64(l * l)) * Float64(1.0 / Float64((abs(k) ^ 4.0) * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(k) * t;
	tmp = 0.0;
	if (abs(k) <= 2.4e+22)
		tmp = (l / ((t_1 * t) * t_1)) * l;
	else
		tmp = (2.0 * (l * l)) * (1.0 / ((abs(k) ^ 4.0) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 2.4e+22], N[(N[(l / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[Power[N[Abs[k], $MachinePrecision], 4.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.4e22

   1. Initial program 54.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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 50.3

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.1

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.7

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lift-*.f64 59.7

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.7%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   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 59.7

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      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. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      13. lower-*.f64 63.2

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 63.2%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lower-*.f64 66.4

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 66.4%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   if 2.4e22 < k

   1. Initial program 54.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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 59.7

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 59.7%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. associate-/l* N/A

         \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \color{blue}{\frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(2 \cdot {\ell}^{2}\right) \cdot \color{blue}{\frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot {\ell}^{2}\right) \cdot \color{blue}{\frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot {\ell}^{2}\right) \cdot \frac{\color{blue}{\cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. pow2 N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      11. lower-/.f64 59.6

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      13. *-commutative N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

      14. lift-pow.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{\color{blue}{2}}} \]

      15. unpow2 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(t \cdot {\sin k}^{2}\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      16. associate-*r* N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot \color{blue}{k}} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot \color{blue}{k}} \]

   6. Applied rewrites 59.0%

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\cos k}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{\color{blue}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{{k}^{4} \cdot \color{blue}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{{k}^{4} \cdot t} \]

      3. lower-pow.f64 51.6

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{{k}^{4} \cdot t} \]

   9. Applied rewrites 51.6%

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{\color{blue}{{k}^{4} \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 67.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} t_1 := k \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{\left|t\right| \cdot \left|t\right|}}{\left|t\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot \left|t\right|\right) \cdot t\_1} \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 4e-5)
      (/ (/ (* 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) <= 4e-5) {
		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) <= 4e-5) {
		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) <= 4e-5:
		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) <= 4e-5)
		tmp = Float64(Float64(Float64(l * Float64(l / Float64(k * k))) / Float64(abs(t) * 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) <= 4e-5)
		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], 4e-5], N[(N[(N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[t], $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 < 4.00000000000000033e-5

   1. Initial program 54.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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 50.3

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{\color{blue}{{t}^{3}}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{\color{blue}{3}}} \]

      5. pow3 N/A

         \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{\left(t \cdot t\right) \cdot \color{blue}{t}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{\left(t \cdot t\right) \cdot t} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t \cdot t}}{\color{blue}{t}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t \cdot t}}{\color{blue}{t}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t \cdot t}}{t} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t \cdot t}}{t} \]

      11. pow2 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot t}}{t} \]

      12. associate-/l* N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t \cdot t}}{t} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t \cdot t}}{t} \]

      14. lower-/.f64 57.5

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t \cdot t}}{t} \]

      15. lift-pow.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t \cdot t}}{t} \]

      16. unpow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t \cdot t}}{t} \]

      17. lower-*.f64 57.5

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t \cdot t}}{t} \]

   6. Applied rewrites 57.5%

      \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t \cdot t}}{\color{blue}{t}} \]

   if 4.00000000000000033e-5 < t

   1. Initial program 54.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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 50.3

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 55.1

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.7

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lift-*.f64 59.7

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.7%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   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 59.7

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      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. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      13. lower-*.f64 63.2

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 63.2%

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lower-*.f64 66.4

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 66.4%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 66.4% 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 54.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. 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 50.3

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 50.3%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
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 55.1

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   8. *-commutative N/A

      \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

   9. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

   10. unpow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

   11. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

   12. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

   13. lower-*.f64 59.7

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   14. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   15. pow3 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   16. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   17. lift-*.f64 59.7

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

6. Applied rewrites 59.7%

   \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
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 59.7

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   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. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   12. *-commutative N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   13. lower-*.f64 63.2

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

8. Applied rewrites 63.2%

   \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   3. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   5. lower-*.f64 66.4

      \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

10. Applied rewrites 66.4%

    \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
11. Add Preprocessing

Alternative 18: 63.2% 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 54.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. 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 50.3

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 50.3%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
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 55.1

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   8. *-commutative N/A

      \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

   9. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

   10. unpow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

   11. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

   12. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

   13. lower-*.f64 59.7

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   14. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   15. pow3 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   16. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   17. lift-*.f64 59.7

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

6. Applied rewrites 59.7%

   \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
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 59.7

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   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. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   12. *-commutative N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   13. lower-*.f64 63.2

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

8. Applied rewrites 63.2%

   \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025172 
(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))))