Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.4% → 90.6%

Time: 7.8s

Alternatives: 17

Speedup: 6.6×

• 72.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.6% accurate, 1.1× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.55 \cdot 10^{-113}:\\ \;\;\;\;\left(\ell + \ell\right) \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left|t\right|\right) \cdot k\right) \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\sin k \cdot \left|t\right|}{\ell} \cdot \left|t\right|\right) \cdot \left(\frac{\left|t\right|}{\ell} \cdot \left(\mathsf{fma}\left(\frac{k}{\left|t\right| \cdot \left|t\right|}, k, 2\right) \cdot \tan k\right)\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 1.55e-113)
    (*
     (+ l l)
     (* l (/ (cos k) (* (* (* (fma (cos (+ k k)) -0.5 0.5) (fabs t)) k) k))))
    (/
     2.0
     (*
      (* (/ (* (sin k) (fabs t)) l) (fabs t))
      (*
       (/ (fabs t) l)
       (* (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) <= 1.55e-113) {
		tmp = (l + l) * (l * (cos(k) / (((fma(cos((k + k)), -0.5, 0.5) * fabs(t)) * k) * k)));
	} else {
		tmp = 2.0 / ((((sin(k) * fabs(t)) / l) * fabs(t)) * ((fabs(t) / l) * (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) <= 1.55e-113)
		tmp = Float64(Float64(l + l) * Float64(l * Float64(cos(k) / Float64(Float64(Float64(fma(cos(Float64(k + k)), -0.5, 0.5) * abs(t)) * k) * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) * abs(t)) / l) * abs(t)) * Float64(Float64(abs(t) / l) * 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], 1.55e-113], N[(N[(l + l), $MachinePrecision] * N[(l * N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[t], $MachinePrecision] / l), $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 < 1.5500000000000001e-113

   1. Initial program 54.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 59.5%

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

   4. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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 59.3%

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

      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 59.2%

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\cos k}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   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 65.1%

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

   8. Applied rewrites 65.1%

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

   if 1.5500000000000001e-113 < t

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 66.4%

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

   3. Applied rewrites 66.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 75.1%

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

   5. Applied rewrites 75.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 70.5%

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

Alternative 2: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\ell}\\ t_2 := \sin k \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.7 \cdot 10^{-113}:\\ \;\;\;\;\left(\ell + \ell\right) \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left|t\right|\right) \cdot k\right) \cdot k}\right)\\ \mathbf{elif}\;\left|t\right| \leq 3 \cdot 10^{+157}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot \left|t\right|\right) \cdot \left(\frac{t\_2}{\ell} \cdot \left(\mathsf{fma}\left(\frac{k}{\left|t\right| \cdot \left|t\right|}, k, 2\right) \cdot \tan k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left|t\right| \cdot \left(\frac{1}{\ell} \cdot \left(\tan k \cdot \left(t\_2 \cdot t\_1\right)\right)\right)\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (fabs t) l)) (t_2 (* (sin k) (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1.7e-113)
      (*
       (+ l l)
       (* l (/ (cos k) (* (* (* (fma (cos (+ k k)) -0.5 0.5) (fabs t)) k) k))))
      (if (<= (fabs t) 3e+157)
        (/
         2.0
         (*
          (* t_1 (fabs t))
          (* (/ t_2 l) (* (fma (/ k (* (fabs t) (fabs t))) k 2.0) (tan k)))))
        (/ 2.0 (* (* (fabs t) (* (/ 1.0 l) (* (tan k) (* t_2 t_1)))) 2.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) / l;
	double t_2 = sin(k) * fabs(t);
	double tmp;
	if (fabs(t) <= 1.7e-113) {
		tmp = (l + l) * (l * (cos(k) / (((fma(cos((k + k)), -0.5, 0.5) * fabs(t)) * k) * k)));
	} else if (fabs(t) <= 3e+157) {
		tmp = 2.0 / ((t_1 * fabs(t)) * ((t_2 / l) * (fma((k / (fabs(t) * fabs(t))), k, 2.0) * tan(k))));
	} else {
		tmp = 2.0 / ((fabs(t) * ((1.0 / l) * (tan(k) * (t_2 * t_1)))) * 2.0);
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) / l)
	t_2 = Float64(sin(k) * abs(t))
	tmp = 0.0
	if (abs(t) <= 1.7e-113)
		tmp = Float64(Float64(l + l) * Float64(l * Float64(cos(k) / Float64(Float64(Float64(fma(cos(Float64(k + k)), -0.5, 0.5) * abs(t)) * k) * k))));
	elseif (abs(t) <= 3e+157)
		tmp = Float64(2.0 / Float64(Float64(t_1 * abs(t)) * Float64(Float64(t_2 / l) * Float64(fma(Float64(k / Float64(abs(t) * abs(t))), k, 2.0) * tan(k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(abs(t) * Float64(Float64(1.0 / l) * Float64(tan(k) * Float64(t_2 * t_1)))) * 2.0));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$2 = 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], 1.7e-113], N[(N[(l + l), $MachinePrecision] * N[(l * N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 3e+157], N[(2.0 / N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 / l), $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], N[(2.0 / N[(N[(N[Abs[t], $MachinePrecision] * N[(N[(1.0 / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.7000000000000001e-113

   1. Initial program 54.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 59.5%

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

   4. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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 59.3%

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

      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 59.2%

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\cos k}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   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 65.1%

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

   8. Applied rewrites 65.1%

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

   if 1.7000000000000001e-113 < t < 3.0000000000000001e157

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      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 66.4%

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

   3. Applied rewrites 66.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 75.1%

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

   5. Applied rewrites 75.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

   7. Applied rewrites 65.4%

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

   if 3.0000000000000001e157 < t

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 66.4%

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

   3. Applied rewrites 66.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 75.1%

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 68.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 69.2%

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lift-/.f64 N/A

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

      18. mult-flip N/A

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

      19. lift-/.f64 N/A

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

   10. Applied rewrites 68.9%

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

Alternative 3: 84.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 8 \cdot 10^{+24}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{1}{\ell} \cdot \left(\tan \left(\left|k\right|\right) \cdot \left(\left(\sin \left(\left|k\right|\right) \cdot t\right) \cdot \frac{t}{\ell}\right)\right)\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell + \ell\right) \cdot \left(\ell \cdot \frac{\cos \left(\left|k\right|\right)}{\left(\left(\mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right) \cdot t\right) \cdot \left|k\right|\right) \cdot \left|k\right|}\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs k) 8e+24)
   (/
    2.0
    (*
     (* t (* (/ 1.0 l) (* (tan (fabs k)) (* (* (sin (fabs k)) t) (/ t l)))))
     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 tmp;
	if (fabs(k) <= 8e+24) {
		tmp = 2.0 / ((t * ((1.0 / l) * (tan(fabs(k)) * ((sin(fabs(k)) * t) * (t / l))))) * 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)
	tmp = 0.0
	if (abs(k) <= 8e+24)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(1.0 / l) * Float64(tan(abs(k)) * Float64(Float64(sin(abs(k)) * t) * Float64(t / l))))) * 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_] := If[LessEqual[N[Abs[k], $MachinePrecision], 8e+24], N[(2.0 / N[(N[(t * N[(N[(1.0 / l), $MachinePrecision] * N[(N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $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 2 regimes

2. if k < 7.9999999999999999e24

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      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 66.4%

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

   3. Applied rewrites 66.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 75.1%

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 68.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 69.2%

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lift-/.f64 N/A

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

      18. mult-flip N/A

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

      19. lift-/.f64 N/A

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

   10. Applied rewrites 68.9%

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

   if 7.9999999999999999e24 < k

   1. Initial program 54.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 59.5%

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

   4. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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 59.3%

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

      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 59.2%

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\cos k}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   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 65.1%

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

   8. Applied rewrites 65.1%

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

Alternative 4: 84.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 1.42 \cdot 10^{+23}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\tan \left(\left|k\right|\right) \cdot \left(\left(\sin \left(\left|k\right|\right) \cdot t\right) \cdot \frac{t}{\ell}\right)\right)}{\ell} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell + \ell\right) \cdot \left(\ell \cdot \frac{\cos \left(\left|k\right|\right)}{\left(\left(\mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right) \cdot t\right) \cdot \left|k\right|\right) \cdot \left|k\right|}\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs k) 1.42e+23)
   (/
    2.0
    (* (/ (* t (* (tan (fabs k)) (* (* (sin (fabs k)) t) (/ t l)))) l) 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 tmp;
	if (fabs(k) <= 1.42e+23) {
		tmp = 2.0 / (((t * (tan(fabs(k)) * ((sin(fabs(k)) * t) * (t / l)))) / l) * 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)
	tmp = 0.0
	if (abs(k) <= 1.42e+23)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(tan(abs(k)) * Float64(Float64(sin(abs(k)) * t) * Float64(t / l)))) / l) * 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_] := If[LessEqual[N[Abs[k], $MachinePrecision], 1.42e+23], N[(2.0 / N[(N[(N[(t * N[(N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $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 2 regimes

2. if k < 1.42e23

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      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 66.4%

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

   3. Applied rewrites 66.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 75.1%

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 68.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   10. Applied rewrites 67.9%

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

   if 1.42e23 < k

   1. Initial program 54.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 59.5%

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

   4. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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 59.3%

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

      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 59.2%

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\cos k}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   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 65.1%

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

   8. Applied rewrites 65.1%

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

Alternative 5: 76.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{t}{\left|\ell\right|}\\ \mathbf{if}\;\left|\ell\right| \leq 0.00043:\\ \;\;\;\;\frac{2}{\left(\left(t\_1 \cdot \left(t \cdot \left(\left(k \cdot t\right) \cdot \frac{1}{\left|\ell\right|}\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\sin k \cdot t\right) \cdot t\_1\right) \cdot t\_1}}{\tan k \cdot 2}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ t (fabs l))))
   (if (<= (fabs l) 0.00043)
     (/
      2.0
      (*
       (* (* t_1 (* t (* (* k t) (/ 1.0 (fabs l))))) (tan k))
       (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
     (/ (/ 2.0 (* (* (* (sin k) t) t_1) t_1)) (* (tan k) 2.0)))))

C

double code(double t, double l, double k) {
	double t_1 = t / fabs(l);
	double tmp;
	if (fabs(l) <= 0.00043) {
		tmp = 2.0 / (((t_1 * (t * ((k * t) * (1.0 / fabs(l))))) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
	} else {
		tmp = (2.0 / (((sin(k) * t) * t_1) * t_1)) / (tan(k) * 2.0);
	}
	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 = t / abs(l)
    if (abs(l) <= 0.00043d0) then
        tmp = 2.0d0 / (((t_1 * (t * ((k * t) * (1.0d0 / abs(l))))) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
    else
        tmp = (2.0d0 / (((sin(k) * t) * t_1) * t_1)) / (tan(k) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = t / Math.abs(l);
	double tmp;
	if (Math.abs(l) <= 0.00043) {
		tmp = 2.0 / (((t_1 * (t * ((k * t) * (1.0 / Math.abs(l))))) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
	} else {
		tmp = (2.0 / (((Math.sin(k) * t) * t_1) * t_1)) / (Math.tan(k) * 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = t / math.fabs(l)
	tmp = 0
	if math.fabs(l) <= 0.00043:
		tmp = 2.0 / (((t_1 * (t * ((k * t) * (1.0 / math.fabs(l))))) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))
	else:
		tmp = (2.0 / (((math.sin(k) * t) * t_1) * t_1)) / (math.tan(k) * 2.0)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(t / abs(l))
	tmp = 0.0
	if (abs(l) <= 0.00043)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_1 * Float64(t * Float64(Float64(k * t) * Float64(1.0 / abs(l))))) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(sin(k) * t) * t_1) * t_1)) / Float64(tan(k) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = t / abs(l);
	tmp = 0.0;
	if (abs(l) <= 0.00043)
		tmp = 2.0 / (((t_1 * (t * ((k * t) * (1.0 / abs(l))))) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
	else
		tmp = (2.0 / (((sin(k) * t) * t_1) * t_1)) / (tan(k) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(t / N[Abs[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[l], $MachinePrecision], 0.00043], N[(2.0 / N[(N[(N[(t$95$1 * N[(t * N[(N[(k * t), $MachinePrecision] * N[(1.0 / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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], N[(N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 4.2999999999999999e-4

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      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 66.4%

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

   3. Applied rewrites 66.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 75.1%

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 69.6%

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

   if 4.2999999999999999e-4 < l

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      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 66.4%

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

   3. Applied rewrites 66.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 75.1%

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 68.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   10. Applied rewrites 68.1%

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

Alternative 6: 76.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{t}{\left|\ell\right|}\\ \mathbf{if}\;\left|\ell\right| \leq 0.00043:\\ \;\;\;\;\frac{2}{\left(\left(t\_1 \cdot \left(t \cdot \left(\left(k \cdot t\right) \cdot \frac{1}{\left|\ell\right|}\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot t\_1\right) \cdot t\_1\right) \cdot \left(\tan k \cdot 2\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ t (fabs l))))
   (if (<= (fabs l) 0.00043)
     (/
      2.0
      (*
       (* (* t_1 (* t (* (* k t) (/ 1.0 (fabs l))))) (tan k))
       (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
     (/ 2.0 (* (* (* (* (sin k) t) t_1) t_1) (* (tan k) 2.0))))))

C

double code(double t, double l, double k) {
	double t_1 = t / fabs(l);
	double tmp;
	if (fabs(l) <= 0.00043) {
		tmp = 2.0 / (((t_1 * (t * ((k * t) * (1.0 / fabs(l))))) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / ((((sin(k) * t) * t_1) * t_1) * (tan(k) * 2.0));
	}
	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 = t / abs(l)
    if (abs(l) <= 0.00043d0) then
        tmp = 2.0d0 / (((t_1 * (t * ((k * t) * (1.0d0 / abs(l))))) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
    else
        tmp = 2.0d0 / ((((sin(k) * t) * t_1) * t_1) * (tan(k) * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = t / Math.abs(l);
	double tmp;
	if (Math.abs(l) <= 0.00043) {
		tmp = 2.0 / (((t_1 * (t * ((k * t) * (1.0 / Math.abs(l))))) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / ((((Math.sin(k) * t) * t_1) * t_1) * (Math.tan(k) * 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = t / math.fabs(l)
	tmp = 0
	if math.fabs(l) <= 0.00043:
		tmp = 2.0 / (((t_1 * (t * ((k * t) * (1.0 / math.fabs(l))))) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))
	else:
		tmp = 2.0 / ((((math.sin(k) * t) * t_1) * t_1) * (math.tan(k) * 2.0))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(t / abs(l))
	tmp = 0.0
	if (abs(l) <= 0.00043)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_1 * Float64(t * Float64(Float64(k * t) * Float64(1.0 / abs(l))))) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) * t) * t_1) * t_1) * Float64(tan(k) * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = t / abs(l);
	tmp = 0.0;
	if (abs(l) <= 0.00043)
		tmp = 2.0 / (((t_1 * (t * ((k * t) * (1.0 / abs(l))))) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
	else
		tmp = 2.0 / ((((sin(k) * t) * t_1) * t_1) * (tan(k) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(t / N[Abs[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[l], $MachinePrecision], 0.00043], N[(2.0 / N[(N[(N[(t$95$1 * N[(t * N[(N[(k * t), $MachinePrecision] * N[(1.0 / N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 4.2999999999999999e-4

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      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 66.4%

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

   3. Applied rewrites 66.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 75.1%

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 69.6%

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

   if 4.2999999999999999e-4 < l

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      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 66.4%

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

   3. Applied rewrites 66.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 75.1%

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 68.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

   10. Applied rewrites 68.1%

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

Alternative 7: 74.3% accurate, 1.4× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 4.2 \cdot 10^{-148}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{4} \cdot \left|t\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{k}{\left|t\right| \cdot \left|t\right|}, 2\right) \cdot \left(\left(\tan k \cdot \frac{\left|t\right|}{\ell}\right) \cdot \left(\frac{k \cdot \left|t\right|}{\ell} \cdot \left|t\right|\right)\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 4.2e-148)
    (* 2.0 (/ (* (pow l 2.0) (cos k)) (* (pow k 4.0) (fabs t))))
    (/
     2.0
     (*
      (fma k (/ k (* (fabs t) (fabs t))) 2.0)
      (* (* (tan k) (/ (fabs t) l)) (* (/ (* k (fabs t)) l) (fabs t))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 4.2e-148) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (pow(k, 4.0) * fabs(t)));
	} else {
		tmp = 2.0 / (fma(k, (k / (fabs(t) * fabs(t))), 2.0) * ((tan(k) * (fabs(t) / l)) * (((k * fabs(t)) / l) * fabs(t))));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 4.2e-148)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64((k ^ 4.0) * abs(t))));
	else
		tmp = Float64(2.0 / Float64(fma(k, Float64(k / Float64(abs(t) * abs(t))), 2.0) * Float64(Float64(tan(k) * Float64(abs(t) / l)) * Float64(Float64(Float64(k * abs(t)) / l) * abs(t)))));
	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], 4.2e-148], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 4.0], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.2e-148

   1. Initial program 54.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 59.5%

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

   4. Applied rewrites 59.5%

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

   5. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{4} \cdot t} \]

      2. lower-pow.f64 51.9%

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{4} \cdot t} \]

   7. Applied rewrites 51.9%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{4} \cdot \color{blue}{t}} \]

   if 4.2e-148 < t

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 66.4%

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

   3. Applied rewrites 66.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 75.1%

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 69.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.6%

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

   10. Applied rewrites 64.0%

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

Alternative 8: 74.1% accurate, 1.4× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 3.8 \cdot 10^{-148}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left({k}^{3} \cdot \left|t\right|\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{k}{\left|t\right| \cdot \left|t\right|}, 2\right) \cdot \left(\left(\tan k \cdot \frac{\left|t\right|}{\ell}\right) \cdot \left(\frac{k \cdot \left|t\right|}{\ell} \cdot \left|t\right|\right)\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 3.8e-148)
    (* (* 2.0 (* l l)) (/ (cos k) (* (* (pow k 3.0) (fabs t)) k)))
    (/
     2.0
     (*
      (fma k (/ k (* (fabs t) (fabs t))) 2.0)
      (* (* (tan k) (/ (fabs t) l)) (* (/ (* k (fabs t)) l) (fabs t))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 3.8e-148) {
		tmp = (2.0 * (l * l)) * (cos(k) / ((pow(k, 3.0) * fabs(t)) * k));
	} else {
		tmp = 2.0 / (fma(k, (k / (fabs(t) * fabs(t))), 2.0) * ((tan(k) * (fabs(t) / l)) * (((k * fabs(t)) / l) * fabs(t))));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 3.8e-148)
		tmp = Float64(Float64(2.0 * Float64(l * l)) * Float64(cos(k) / Float64(Float64((k ^ 3.0) * abs(t)) * k)));
	else
		tmp = Float64(2.0 / Float64(fma(k, Float64(k / Float64(abs(t) * abs(t))), 2.0) * Float64(Float64(tan(k) * Float64(abs(t) / l)) * Float64(Float64(Float64(k * abs(t)) / l) * abs(t)))));
	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], 3.8e-148], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[(N[Power[k, 3.0], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.8000000000000001e-148

   1. Initial program 54.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 59.5%

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

   4. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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 59.3%

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

      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 59.2%

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

   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 52.6%

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

   9. Applied rewrites 52.6%

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

   if 3.8000000000000001e-148 < t

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 66.4%

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

   3. Applied rewrites 66.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 75.1%

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 69.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.6%

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

   10. Applied rewrites 64.0%

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

Alternative 9: 70.3% accurate, 1.6× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 6.5 \cdot 10^{-101}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left({k}^{3} \cdot \left|t\right|\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left|t\right|}{\ell} \cdot \left(\left|t\right| \cdot \frac{k \cdot \left|t\right|}{\ell}\right)\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 6.5e-101)
    (* (* 2.0 (* l l)) (/ (cos k) (* (* (pow k 3.0) (fabs t)) k)))
    (/
     2.0
     (*
      (* (* (/ (fabs t) l) (* (fabs t) (/ (* k (fabs t)) l))) (tan k))
      2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 6.5e-101) {
		tmp = (2.0 * (l * l)) * (cos(k) / ((pow(k, 3.0) * fabs(t)) * k));
	} else {
		tmp = 2.0 / ((((fabs(t) / l) * (fabs(t) * ((k * fabs(t)) / l))) * tan(k)) * 2.0);
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(t) <= 6.5e-101) {
		tmp = (2.0 * (l * l)) * (Math.cos(k) / ((Math.pow(k, 3.0) * Math.abs(t)) * k));
	} else {
		tmp = 2.0 / ((((Math.abs(t) / l) * (Math.abs(t) * ((k * Math.abs(t)) / l))) * Math.tan(k)) * 2.0);
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(t) <= 6.5e-101:
		tmp = (2.0 * (l * l)) * (math.cos(k) / ((math.pow(k, 3.0) * math.fabs(t)) * k))
	else:
		tmp = 2.0 / ((((math.fabs(t) / l) * (math.fabs(t) * ((k * math.fabs(t)) / l))) * math.tan(k)) * 2.0)
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 6.5e-101)
		tmp = Float64(Float64(2.0 * Float64(l * l)) * Float64(cos(k) / Float64(Float64((k ^ 3.0) * abs(t)) * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(abs(t) / l) * Float64(abs(t) * Float64(Float64(k * abs(t)) / l))) * tan(k)) * 2.0));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(t) <= 6.5e-101)
		tmp = (2.0 * (l * l)) * (cos(k) / (((k ^ 3.0) * abs(t)) * k));
	else
		tmp = 2.0 / ((((abs(t) / l) * (abs(t) * ((k * abs(t)) / l))) * tan(k)) * 2.0);
	end
	tmp_2 = (sign(t) * abs(1.0)) * 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], 6.5e-101], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[(N[Power[k, 3.0], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.4999999999999996e-101

   1. Initial program 54.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 59.5%

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

   4. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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 59.3%

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

      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 59.2%

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

   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 52.6%

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

   9. Applied rewrites 52.6%

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

   if 6.4999999999999996e-101 < t

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 66.4%

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

   3. Applied rewrites 66.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 75.1%

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 68.1%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 65.9%

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

   11. Applied rewrites 65.9%

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

Alternative 10: 69.1% accurate, 1.7× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 6.5 \cdot 10^{-101}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{{k}^{4} \cdot \left|t\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left|t\right|}{\ell} \cdot \left(\left|t\right| \cdot \frac{k \cdot \left|t\right|}{\ell}\right)\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 6.5e-101)
    (* (* 2.0 (* l l)) (/ (cos k) (* (pow k 4.0) (fabs t))))
    (/
     2.0
     (*
      (* (* (/ (fabs t) l) (* (fabs t) (/ (* k (fabs t)) l))) (tan k))
      2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 6.5e-101) {
		tmp = (2.0 * (l * l)) * (cos(k) / (pow(k, 4.0) * fabs(t)));
	} else {
		tmp = 2.0 / ((((fabs(t) / l) * (fabs(t) * ((k * fabs(t)) / l))) * tan(k)) * 2.0);
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(t) <= 6.5e-101) {
		tmp = (2.0 * (l * l)) * (Math.cos(k) / (Math.pow(k, 4.0) * Math.abs(t)));
	} else {
		tmp = 2.0 / ((((Math.abs(t) / l) * (Math.abs(t) * ((k * Math.abs(t)) / l))) * Math.tan(k)) * 2.0);
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(t) <= 6.5e-101:
		tmp = (2.0 * (l * l)) * (math.cos(k) / (math.pow(k, 4.0) * math.fabs(t)))
	else:
		tmp = 2.0 / ((((math.fabs(t) / l) * (math.fabs(t) * ((k * math.fabs(t)) / l))) * math.tan(k)) * 2.0)
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 6.5e-101)
		tmp = Float64(Float64(2.0 * Float64(l * l)) * Float64(cos(k) / Float64((k ^ 4.0) * abs(t))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(abs(t) / l) * Float64(abs(t) * Float64(Float64(k * abs(t)) / l))) * tan(k)) * 2.0));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(t) <= 6.5e-101)
		tmp = (2.0 * (l * l)) * (cos(k) / ((k ^ 4.0) * abs(t)));
	else
		tmp = 2.0 / ((((abs(t) / l) * (abs(t) * ((k * abs(t)) / l))) * tan(k)) * 2.0);
	end
	tmp_2 = (sign(t) * abs(1.0)) * 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], 6.5e-101], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[Power[k, 4.0], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.4999999999999996e-101

   1. Initial program 54.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 59.5%

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

   4. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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 59.3%

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

      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 59.2%

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

   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 51.9%

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

   9. Applied rewrites 51.9%

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

   if 6.4999999999999996e-101 < t

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 66.4%

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

   3. Applied rewrites 66.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 75.1%

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 68.1%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 65.9%

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

   11. Applied rewrites 65.9%

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

Alternative 11: 68.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 5.1 \cdot 10^{+115}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\left|k\right| \cdot t}{\ell}\right)\right) \cdot \tan \left(\left|k\right|\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\left(\left|k\right| \cdot \left|k\right|\right) \cdot t}}{t}}{t} \cdot \ell\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs k) 5.1e+115)
   (/ 2.0 (* (* (* (/ t l) (* t (/ (* (fabs k) t) l))) (tan (fabs k))) 2.0))
   (* (/ (/ (/ l (* (* (fabs k) (fabs k)) t)) t) t) l)))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 5.1e+115) {
		tmp = 2.0 / ((((t / l) * (t * ((fabs(k) * t) / l))) * tan(fabs(k))) * 2.0);
	} else {
		tmp = (((l / ((fabs(k) * fabs(k)) * t)) / t) / t) * l;
	}
	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) :: tmp
    if (abs(k) <= 5.1d+115) then
        tmp = 2.0d0 / ((((t / l) * (t * ((abs(k) * t) / l))) * tan(abs(k))) * 2.0d0)
    else
        tmp = (((l / ((abs(k) * abs(k)) * t)) / t) / t) * l
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(k) <= 5.1e+115) {
		tmp = 2.0 / ((((t / l) * (t * ((Math.abs(k) * t) / l))) * Math.tan(Math.abs(k))) * 2.0);
	} else {
		tmp = (((l / ((Math.abs(k) * Math.abs(k)) * t)) / t) / t) * l;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(k) <= 5.1e+115:
		tmp = 2.0 / ((((t / l) * (t * ((math.fabs(k) * t) / l))) * math.tan(math.fabs(k))) * 2.0)
	else:
		tmp = (((l / ((math.fabs(k) * math.fabs(k)) * t)) / t) / t) * l
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 5.1e+115)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t / l) * Float64(t * Float64(Float64(abs(k) * t) / l))) * tan(abs(k))) * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(l / Float64(Float64(abs(k) * abs(k)) * t)) / t) / t) * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(k) <= 5.1e+115)
		tmp = 2.0 / ((((t / l) * (t * ((abs(k) * t) / l))) * tan(abs(k))) * 2.0);
	else
		tmp = (((l / ((abs(k) * abs(k)) * t)) / t) / t) * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 5.1e+115], N[(2.0 / N[(N[(N[(N[(t / l), $MachinePrecision] * N[(t * N[(N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / N[(N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / t), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.0999999999999996e115

   1. Initial program 54.4%

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

      1. lift-*.f64 N/A

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

      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 66.4%

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

   3. Applied rewrites 66.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 75.1%

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 68.1%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 65.9%

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

   11. Applied rewrites 65.9%

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

   if 5.0999999999999996e115 < k

   1. Initial program 54.4%

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

   2. Taylor expanded in 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 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 54.9%

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

      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 59.5%

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

      14. lift-pow.f64 N/A

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

      15. unpow3 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. lower-*.f64 59.5%

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

   6. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.5%

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow3 N/A

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

      9. lift-pow.f64 N/A

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

      10. associate-*l* N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. cube-mult N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 57.7%

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

      20. lift-pow.f64 N/A

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

      21. unpow2 N/A

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

      22. lower-*.f64 57.7%

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

   8. Applied rewrites 57.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 62.7%

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

   10. Applied rewrites 62.7%

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

Alternative 12: 67.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 1.15 \cdot 10^{-161}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot \left(\left|k\right| \cdot t\right)\right)\right) \cdot \left|k\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\left(\left|k\right| \cdot \left|k\right|\right) \cdot t}}{t}}{t} \cdot \ell\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs k) 1.15e-161)
   (* l (/ l (* (* t (* t (* (fabs k) t))) (fabs k))))
   (* (/ (/ (/ l (* (* (fabs k) (fabs k)) t)) t) t) l)))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 1.15e-161) {
		tmp = l * (l / ((t * (t * (fabs(k) * t))) * fabs(k)));
	} else {
		tmp = (((l / ((fabs(k) * fabs(k)) * t)) / t) / t) * l;
	}
	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) :: tmp
    if (abs(k) <= 1.15d-161) then
        tmp = l * (l / ((t * (t * (abs(k) * t))) * abs(k)))
    else
        tmp = (((l / ((abs(k) * abs(k)) * t)) / t) / t) * l
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(k) <= 1.15e-161) {
		tmp = l * (l / ((t * (t * (Math.abs(k) * t))) * Math.abs(k)));
	} else {
		tmp = (((l / ((Math.abs(k) * Math.abs(k)) * t)) / t) / t) * l;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(k) <= 1.15e-161:
		tmp = l * (l / ((t * (t * (math.fabs(k) * t))) * math.fabs(k)))
	else:
		tmp = (((l / ((math.fabs(k) * math.fabs(k)) * t)) / t) / t) * l
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 1.15e-161)
		tmp = Float64(l * Float64(l / Float64(Float64(t * Float64(t * Float64(abs(k) * t))) * abs(k))));
	else
		tmp = Float64(Float64(Float64(Float64(l / Float64(Float64(abs(k) * abs(k)) * t)) / t) / t) * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(k) <= 1.15e-161)
		tmp = l * (l / ((t * (t * (abs(k) * t))) * abs(k)));
	else
		tmp = (((l / ((abs(k) * abs(k)) * t)) / t) / t) * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 1.15e-161], N[(l * N[(l / N[(N[(t * N[(t * N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / N[(N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / t), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.15e-161

   1. Initial program 54.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 54.9%

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

      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 59.5%

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

      14. lift-pow.f64 N/A

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

      15. unpow3 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. lower-*.f64 59.5%

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

   6. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 63.5%

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

   8. Applied rewrites 63.5%

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

   if 1.15e-161 < k

   1. Initial program 54.4%

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

   2. Taylor expanded in 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 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 54.9%

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

      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 59.5%

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

      14. lift-pow.f64 N/A

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

      15. unpow3 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. lower-*.f64 59.5%

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

   6. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.5%

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow3 N/A

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

      9. lift-pow.f64 N/A

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

      10. associate-*l* N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. cube-mult N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 57.7%

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

      20. lift-pow.f64 N/A

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

      21. unpow2 N/A

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

      22. lower-*.f64 57.7%

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

   8. Applied rewrites 57.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 62.7%

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

   10. Applied rewrites 62.7%

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

Alternative 13: 67.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 5.2 \cdot 10^{-162}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot \left(\left|k\right| \cdot t\right)\right)\right) \cdot \left|k\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(\left|k\right| \cdot \left|k\right|\right) \cdot t\right) \cdot t} \cdot \frac{\ell}{t}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs k) 5.2e-162)
   (* l (/ l (* (* t (* t (* (fabs k) t))) (fabs k))))
   (* (/ l (* (* (* (fabs k) (fabs k)) t) t)) (/ l t))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 5.2e-162) {
		tmp = l * (l / ((t * (t * (fabs(k) * t))) * fabs(k)));
	} else {
		tmp = (l / (((fabs(k) * fabs(k)) * t) * t)) * (l / 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) :: tmp
    if (abs(k) <= 5.2d-162) then
        tmp = l * (l / ((t * (t * (abs(k) * t))) * abs(k)))
    else
        tmp = (l / (((abs(k) * abs(k)) * t) * t)) * (l / t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(k) <= 5.2e-162) {
		tmp = l * (l / ((t * (t * (Math.abs(k) * t))) * Math.abs(k)));
	} else {
		tmp = (l / (((Math.abs(k) * Math.abs(k)) * t) * t)) * (l / t);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(k) <= 5.2e-162:
		tmp = l * (l / ((t * (t * (math.fabs(k) * t))) * math.fabs(k)))
	else:
		tmp = (l / (((math.fabs(k) * math.fabs(k)) * t) * t)) * (l / t)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 5.2e-162)
		tmp = Float64(l * Float64(l / Float64(Float64(t * Float64(t * Float64(abs(k) * t))) * abs(k))));
	else
		tmp = Float64(Float64(l / Float64(Float64(Float64(abs(k) * abs(k)) * t) * t)) * Float64(l / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(k) <= 5.2e-162)
		tmp = l * (l / ((t * (t * (abs(k) * t))) * abs(k)));
	else
		tmp = (l / (((abs(k) * abs(k)) * t) * t)) * (l / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 5.2e-162], N[(l * N[(l / N[(N[(t * N[(t * N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.1999999999999999e-162

   1. Initial program 54.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 54.9%

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

      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 59.5%

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

      14. lift-pow.f64 N/A

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

      15. unpow3 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. lower-*.f64 59.5%

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

   6. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 63.5%

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

   8. Applied rewrites 63.5%

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

   if 5.1999999999999999e-162 < k

   1. Initial program 54.4%

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

   2. Taylor expanded in 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 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 54.9%

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

      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 59.5%

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

      14. lift-pow.f64 N/A

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

      15. unpow3 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. lower-*.f64 59.5%

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

   6. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.5%

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow3 N/A

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

      9. lift-pow.f64 N/A

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

      10. associate-*l* N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. cube-mult N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 57.7%

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

      20. lift-pow.f64 N/A

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

      21. unpow2 N/A

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

      22. lower-*.f64 57.7%

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

   8. Applied rewrites 57.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 62.7%

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

   10. Applied rewrites 62.7%

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

Alternative 14: 65.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 7.8 \cdot 10^{+186}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot \left(\left|k\right| \cdot t\right)\right)\right) \cdot \left|k\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(\left(\left|k\right| \cdot \left|k\right|\right) \cdot t\right) \cdot t\right) \cdot t} \cdot \ell\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs k) 7.8e+186)
   (* l (/ l (* (* t (* t (* (fabs k) t))) (fabs k))))
   (* (/ l (* (* (* (* (fabs k) (fabs k)) t) t) t)) l)))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 7.8e+186) {
		tmp = l * (l / ((t * (t * (fabs(k) * t))) * fabs(k)));
	} else {
		tmp = (l / ((((fabs(k) * fabs(k)) * t) * t) * t)) * l;
	}
	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) :: tmp
    if (abs(k) <= 7.8d+186) then
        tmp = l * (l / ((t * (t * (abs(k) * t))) * abs(k)))
    else
        tmp = (l / ((((abs(k) * abs(k)) * t) * t) * t)) * l
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(k) <= 7.8e+186) {
		tmp = l * (l / ((t * (t * (Math.abs(k) * t))) * Math.abs(k)));
	} else {
		tmp = (l / ((((Math.abs(k) * Math.abs(k)) * t) * t) * t)) * l;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(k) <= 7.8e+186:
		tmp = l * (l / ((t * (t * (math.fabs(k) * t))) * math.fabs(k)))
	else:
		tmp = (l / ((((math.fabs(k) * math.fabs(k)) * t) * t) * t)) * l
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 7.8e+186)
		tmp = Float64(l * Float64(l / Float64(Float64(t * Float64(t * Float64(abs(k) * t))) * abs(k))));
	else
		tmp = Float64(Float64(l / Float64(Float64(Float64(Float64(abs(k) * abs(k)) * t) * t) * t)) * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(k) <= 7.8e+186)
		tmp = l * (l / ((t * (t * (abs(k) * t))) * abs(k)));
	else
		tmp = (l / ((((abs(k) * abs(k)) * t) * t) * t)) * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 7.8e+186], N[(l * N[(l / N[(N[(t * N[(t * N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(N[(N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.8000000000000002e186

   1. Initial program 54.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 54.9%

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

      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 59.5%

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

      14. lift-pow.f64 N/A

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

      15. unpow3 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. lower-*.f64 59.5%

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

   6. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 63.5%

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

   8. Applied rewrites 63.5%

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

   if 7.8000000000000002e186 < k

   1. Initial program 54.4%

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

   2. Taylor expanded in 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 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 54.9%

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

      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 59.5%

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

      14. lift-pow.f64 N/A

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

      15. unpow3 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. lower-*.f64 59.5%

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

   6. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.5%

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow3 N/A

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

      9. lift-pow.f64 N/A

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

      10. associate-*l* N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. cube-mult N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 57.7%

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

      20. lift-pow.f64 N/A

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

      21. unpow2 N/A

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

      22. lower-*.f64 57.7%

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

   8. Applied rewrites 57.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 61.5%

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

   10. Applied rewrites 61.5%

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

Alternative 15: 65.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \leq \infty:\\ \;\;\;\;\frac{\ell}{\left(\left(k \cdot \left(t \cdot t\right)\right) \cdot t\right) \cdot k} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot t} \cdot \ell\\ \end{array} \]

FPCore

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

C

double code(double t, double l, double k) {
	double tmp;
	if (((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0)) <= ((double) INFINITY)) {
		tmp = (l / (((k * (t * t)) * t) * k)) * l;
	} else {
		tmp = (l / ((((k * k) * t) * t) * t)) * l;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0)) <= Double.POSITIVE_INFINITY) {
		tmp = (l / (((k * (t * t)) * t) * k)) * l;
	} else {
		tmp = (l / ((((k * k) * t) * t) * t)) * l;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0)) <= math.inf:
		tmp = (l / (((k * (t * t)) * t) * k)) * l
	else:
		tmp = (l / ((((k * k) * t) * t) * t)) * l
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (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)) <= Inf)
		tmp = Float64(Float64(l / Float64(Float64(Float64(k * Float64(t * t)) * t) * k)) * l);
	else
		tmp = Float64(Float64(l / Float64(Float64(Float64(Float64(k * k) * t) * t) * t)) * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0)) <= Inf)
		tmp = (l / (((k * (t * t)) * t) * k)) * l;
	else
		tmp = (l / ((((k * k) * t) * t) * t)) * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[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], Infinity], N[(N[(l / N[(N[(N[(k * N[(t * t), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(l / N[(N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 54.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 54.9%

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

      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 59.5%

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

      14. lift-pow.f64 N/A

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

      15. unpow3 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. lower-*.f64 59.5%

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

   6. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.5%

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow3 N/A

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

      9. lift-pow.f64 N/A

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

      10. associate-*l* N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. cube-mult N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 57.7%

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

      20. lift-pow.f64 N/A

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

      21. unpow2 N/A

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

      22. lower-*.f64 57.7%

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

   8. Applied rewrites 57.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. exp-to-pow N/A

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

      9. lift-log.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 30.0%

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

   10. Applied rewrites 62.4%

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

   if +inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))

   1. Initial program 54.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 50.6%

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 54.9%

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

      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 59.5%

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

      14. lift-pow.f64 N/A

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

      15. unpow3 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. lower-*.f64 59.5%

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

   6. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.5%

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow3 N/A

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

      9. lift-pow.f64 N/A

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

      10. associate-*l* N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. cube-mult N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 57.7%

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

      20. lift-pow.f64 N/A

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

      21. unpow2 N/A

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

      22. lower-*.f64 57.7%

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

   8. Applied rewrites 57.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 61.5%

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

   10. Applied rewrites 61.5%

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

Alternative 16: 62.4% accurate, 6.6× speedup

Math

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

FPCore

(FPCore (t l k) :precision binary64 (* (/ l (* (* (* k (* t t)) t) k)) l))

C

double code(double t, double l, double k) {
	return (l / (((k * (t * t)) * t) * k)) * 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)) * t) * k)) * l
end function

Java

public static double code(double t, double l, double k) {
	return (l / (((k * (t * t)) * t) * k)) * l;
}

Python

def code(t, l, k):
	return (l / (((k * (t * t)) * t) * k)) * l

Julia

function code(t, l, k)
	return Float64(Float64(l / Float64(Float64(Float64(k * Float64(t * t)) * t) * k)) * l)
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / (((k * (t * t)) * t) * k)) * l;
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[(N[(N[(k * N[(t * t), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]

Derivation

1. Initial program 54.4%

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

2. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 50.6%

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

4. Applied rewrites 50.6%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
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 54.9%

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

   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 59.5%

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

   14. lift-pow.f64 N/A

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

   15. unpow3 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. lower-*.f64 59.5%

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

6. Applied rewrites 59.5%

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

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   3. lower-*.f64 59.5%

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   8. pow3 N/A

      \[\leadsto \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \cdot \ell \]

   9. lift-pow.f64 N/A

      \[\leadsto \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \cdot \ell \]

   10. associate-*l* N/A

       \[\leadsto \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)} \cdot \ell \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell}{{t}^{3} \cdot {k}^{2}} \cdot \ell \]

   12. lift-pow.f64 N/A

       \[\leadsto \frac{\ell}{{t}^{3} \cdot {k}^{2}} \cdot \ell \]

   13. *-commutative N/A

       \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{3}} \cdot \ell \]

   14. lift-pow.f64 N/A

       \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{3}} \cdot \ell \]

   15. cube-mult N/A

       \[\leadsto \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

   17. associate-*r* N/A

       \[\leadsto \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

   18. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

   19. lower-*.f64 57.7%

       \[\leadsto \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

   20. lift-pow.f64 N/A

       \[\leadsto \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

   21. unpow2 N/A

       \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

   22. lower-*.f64 57.7%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

8. Applied rewrites 57.7%

   \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

   5. associate-*l* N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \ell \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \ell \]

   7. pow2 N/A

      \[\leadsto \frac{\ell}{\left({t}^{2} \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \ell \]

   8. exp-to-pow N/A

      \[\leadsto \frac{\ell}{\left(e^{\log t \cdot 2} \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \ell \]

   9. lift-log.f64 N/A

      \[\leadsto \frac{\ell}{\left(e^{\log t \cdot 2} \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \ell \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(e^{\log t \cdot 2} \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \ell \]

   11. lift-exp.f64 N/A

       \[\leadsto \frac{\ell}{\left(e^{\log t \cdot 2} \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \ell \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(e^{\log t \cdot 2} \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \ell \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(e^{\log t \cdot 2} \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \ell \]

   14. associate-*l* N/A

       \[\leadsto \frac{\ell}{\left(\left(e^{\log t \cdot 2} \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(\left(e^{\log t \cdot 2} \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   16. lift-*.f64 30.0%

       \[\leadsto \frac{\ell}{\left(\left(e^{\log t \cdot 2} \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

10. Applied rewrites 62.4%

    \[\leadsto \frac{\ell}{\left(\left(k \cdot \left(t \cdot t\right)\right) \cdot t\right) \cdot k} \cdot \ell \]
11. Add Preprocessing

Alternative 17: 57.7% accurate, 6.6× speedup

Math

\[\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ l (* (* (* k k) t) (* t t))) l))

C

double code(double t, double l, double k) {
	return (l / (((k * k) * t) * (t * t))) * l;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l / (((k * k) * t) * (t * t))) * l
end function

Java

public static double code(double t, double l, double k) {
	return (l / (((k * k) * t) * (t * t))) * l;
}

Python

def code(t, l, k):
	return (l / (((k * k) * t) * (t * t))) * l

Julia

function code(t, l, k)
	return Float64(Float64(l / Float64(Float64(Float64(k * k) * t) * Float64(t * t))) * l)
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / (((k * k) * t) * (t * t))) * l;
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]

Derivation

1. Initial program 54.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   4. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   5. lower-pow.f64 50.6%

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 50.6%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
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 54.9%

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   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 59.5%

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   14. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   15. unpow3 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. lower-*.f64 59.5%

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

6. Applied rewrites 59.5%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   3. lower-*.f64 59.5%

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   8. pow3 N/A

      \[\leadsto \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \cdot \ell \]

   9. lift-pow.f64 N/A

      \[\leadsto \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \cdot \ell \]

   10. associate-*l* N/A

       \[\leadsto \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)} \cdot \ell \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell}{{t}^{3} \cdot {k}^{2}} \cdot \ell \]

   12. lift-pow.f64 N/A

       \[\leadsto \frac{\ell}{{t}^{3} \cdot {k}^{2}} \cdot \ell \]

   13. *-commutative N/A

       \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{3}} \cdot \ell \]

   14. lift-pow.f64 N/A

       \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{3}} \cdot \ell \]

   15. cube-mult N/A

       \[\leadsto \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

   17. associate-*r* N/A

       \[\leadsto \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

   18. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

   19. lower-*.f64 57.7%

       \[\leadsto \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

   20. lift-pow.f64 N/A

       \[\leadsto \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

   21. unpow2 N/A

       \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

   22. lower-*.f64 57.7%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

8. Applied rewrites 57.7%

   \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025187 
(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))))