Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.5% → 93.1%

Time: 7.9s

Alternatives: 12

Speedup: 4.4×

• 39.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.5% 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: 93.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \cos \left(\left|k\right|\right)\\ \mathbf{if}\;\left|k\right| \leq 4 \cdot 10^{+73}:\\ \;\;\;\;\frac{\frac{\left(\ell + \ell\right) \cdot t\_1}{{\sin \left(\left|k\right|\right)}^{2} \cdot t}}{\left|k\right| \cdot \left|k\right|} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\frac{t\_1 \cdot \ell}{\left|k\right|}}{\mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right) \cdot t}}{\left|k\right|}\right) \cdot \ell\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cos (fabs k))))
   (if (<= (fabs k) 4e+73)
     (*
      (/
       (/ (* (+ l l) t_1) (* (pow (sin (fabs k)) 2.0) t))
       (* (fabs k) (fabs k)))
      l)
     (*
      (*
       2.0
       (/
        (/
         (/ (* t_1 l) (fabs k))
         (* (fma (cos (+ (fabs k) (fabs k))) -0.5 0.5) t))
        (fabs k)))
      l))))

C

double code(double t, double l, double k) {
	double t_1 = cos(fabs(k));
	double tmp;
	if (fabs(k) <= 4e+73) {
		tmp = ((((l + l) * t_1) / (pow(sin(fabs(k)), 2.0) * t)) / (fabs(k) * fabs(k))) * l;
	} else {
		tmp = (2.0 * ((((t_1 * l) / fabs(k)) / (fma(cos((fabs(k) + fabs(k))), -0.5, 0.5) * t)) / fabs(k))) * l;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cos(abs(k))
	tmp = 0.0
	if (abs(k) <= 4e+73)
		tmp = Float64(Float64(Float64(Float64(Float64(l + l) * t_1) / Float64((sin(abs(k)) ^ 2.0) * t)) / Float64(abs(k) * abs(k))) * l);
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(Float64(Float64(t_1 * l) / abs(k)) / Float64(fma(cos(Float64(abs(k) + abs(k))), -0.5, 0.5) * t)) / abs(k))) * l);
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 4e+73], N[(N[(N[(N[(N[(l + l), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[Power[N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(2.0 * N[(N[(N[(N[(t$95$1 * l), $MachinePrecision] / N[Abs[k], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.99999999999999993e73

   1. Initial program 35.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 37.6%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 75.8%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. count-2 N/A

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

      9. sqr-sin-a N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 85.6

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

   10. Applied rewrites 85.6%

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

   if 3.99999999999999993e73 < k

   1. Initial program 35.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 37.6%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. sqr-sin-a-rev N/A

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

      11. count-2 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

   8. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 83.4

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

   10. Applied rewrites 83.4%

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

Alternative 2: 93.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{{\left(\left|k\right|\right)}^{2} \cdot t}}{\left|k\right| \cdot \left|k\right|} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\frac{\cos \left(\left|k\right|\right) \cdot \ell}{\left|k\right|}}{\mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right) \cdot t}}{\left|k\right|}\right) \cdot \ell\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs k) 2.7e-5)
   (* (/ (* 2.0 (/ l (* (pow (fabs k) 2.0) t))) (* (fabs k) (fabs k))) l)
   (*
    (*
     2.0
     (/
      (/
       (/ (* (cos (fabs k)) l) (fabs k))
       (* (fma (cos (+ (fabs k) (fabs k))) -0.5 0.5) t))
      (fabs k)))
    l)))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 2.7e-5) {
		tmp = ((2.0 * (l / (pow(fabs(k), 2.0) * t))) / (fabs(k) * fabs(k))) * l;
	} else {
		tmp = (2.0 * ((((cos(fabs(k)) * l) / fabs(k)) / (fma(cos((fabs(k) + fabs(k))), -0.5, 0.5) * t)) / fabs(k))) * l;
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 2.7e-5)
		tmp = Float64(Float64(Float64(2.0 * Float64(l / Float64((abs(k) ^ 2.0) * t))) / Float64(abs(k) * abs(k))) * l);
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(Float64(Float64(cos(abs(k)) * l) / abs(k)) / Float64(fma(cos(Float64(abs(k) + abs(k))), -0.5, 0.5) * t)) / abs(k))) * l);
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 2.7e-5], N[(N[(N[(2.0 * N[(l / N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(2.0 * N[(N[(N[(N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] / N[Abs[k], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.6999999999999999e-5

   1. Initial program 35.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 37.6%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 75.8%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 72.8

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

   11. Applied rewrites 72.8%

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

   if 2.6999999999999999e-5 < k

   1. Initial program 35.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 37.6%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. sqr-sin-a-rev N/A

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

      11. count-2 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

   8. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 83.4

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

   10. Applied rewrites 83.4%

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

Alternative 3: 91.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{{\left(\left|k\right|\right)}^{2} \cdot t}}{\left|k\right| \cdot \left|k\right|} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\ell + \ell\right) \cdot \cos \left(\left|k\right|\right)}{\left(\mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right) \cdot t\right) \cdot \left|k\right|}}{\left|k\right|} \cdot \ell\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs k) 2.7e-5)
   (* (/ (* 2.0 (/ l (* (pow (fabs k) 2.0) t))) (* (fabs k) (fabs k))) l)
   (*
    (/
     (/
      (* (+ l l) (cos (fabs k)))
      (* (* (fma (cos (+ (fabs k) (fabs k))) -0.5 0.5) t) (fabs k)))
     (fabs k))
    l)))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 2.7e-5) {
		tmp = ((2.0 * (l / (pow(fabs(k), 2.0) * t))) / (fabs(k) * fabs(k))) * l;
	} else {
		tmp = ((((l + l) * cos(fabs(k))) / ((fma(cos((fabs(k) + fabs(k))), -0.5, 0.5) * t) * fabs(k))) / fabs(k)) * l;
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 2.7e-5)
		tmp = Float64(Float64(Float64(2.0 * Float64(l / Float64((abs(k) ^ 2.0) * t))) / Float64(abs(k) * abs(k))) * l);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(l + l) * cos(abs(k))) / Float64(Float64(fma(cos(Float64(abs(k) + abs(k))), -0.5, 0.5) * t) * abs(k))) / abs(k)) * l);
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 2.7e-5], N[(N[(N[(2.0 * N[(l / N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(N[(N[(N[(l + l), $MachinePrecision] * N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[k], $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.6999999999999999e-5

   1. Initial program 35.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 37.6%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 75.8%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 72.8

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

   11. Applied rewrites 72.8%

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

   if 2.6999999999999999e-5 < k

   1. Initial program 35.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 37.6%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. sqr-sin-a-rev N/A

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

      11. count-2 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

   8. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. count-2 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-/.f64 82.3

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

   10. Applied rewrites 82.3%

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

Alternative 4: 89.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \left|k\right| \cdot \left|k\right|\\ t_2 := \cos \left(\left|k\right|\right)\\ t_3 := \mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right) \cdot t\\ \mathbf{if}\;\left|k\right| \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{{\left(\left|k\right|\right)}^{2} \cdot t}}{t\_1} \cdot \ell\\ \mathbf{elif}\;\left|k\right| \leq 1.7 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{t\_2}{t\_3} \cdot \left(\ell + \ell\right)}{t\_1} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t\_2}{\left(t\_3 \cdot \left|k\right|\right) \cdot \left|k\right|} \cdot \left(\ell + \ell\right)\right) \cdot \ell\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (fabs k) (fabs k)))
        (t_2 (cos (fabs k)))
        (t_3 (* (fma (cos (+ (fabs k) (fabs k))) -0.5 0.5) t)))
   (if (<= (fabs k) 2.7e-5)
     (* (/ (* 2.0 (/ l (* (pow (fabs k) 2.0) t))) t_1) l)
     (if (<= (fabs k) 1.7e+155)
       (* (/ (* (/ t_2 t_3) (+ l l)) t_1) l)
       (* (* (/ t_2 (* (* t_3 (fabs k)) (fabs k))) (+ l l)) l)))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(k) * fabs(k);
	double t_2 = cos(fabs(k));
	double t_3 = fma(cos((fabs(k) + fabs(k))), -0.5, 0.5) * t;
	double tmp;
	if (fabs(k) <= 2.7e-5) {
		tmp = ((2.0 * (l / (pow(fabs(k), 2.0) * t))) / t_1) * l;
	} else if (fabs(k) <= 1.7e+155) {
		tmp = (((t_2 / t_3) * (l + l)) / t_1) * l;
	} else {
		tmp = ((t_2 / ((t_3 * fabs(k)) * fabs(k))) * (l + l)) * l;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(abs(k) * abs(k))
	t_2 = cos(abs(k))
	t_3 = Float64(fma(cos(Float64(abs(k) + abs(k))), -0.5, 0.5) * t)
	tmp = 0.0
	if (abs(k) <= 2.7e-5)
		tmp = Float64(Float64(Float64(2.0 * Float64(l / Float64((abs(k) ^ 2.0) * t))) / t_1) * l);
	elseif (abs(k) <= 1.7e+155)
		tmp = Float64(Float64(Float64(Float64(t_2 / t_3) * Float64(l + l)) / t_1) * l);
	else
		tmp = Float64(Float64(Float64(t_2 / Float64(Float64(t_3 * abs(k)) * abs(k))) * Float64(l + l)) * l);
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 2.7e-5], N[(N[(N[(2.0 * N[(l / N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * l), $MachinePrecision], If[LessEqual[N[Abs[k], $MachinePrecision], 1.7e+155], N[(N[(N[(N[(t$95$2 / t$95$3), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * l), $MachinePrecision], N[(N[(N[(t$95$2 / N[(N[(t$95$3 * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.6999999999999999e-5

   1. Initial program 35.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 37.6%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 75.8%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 72.8

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

   11. Applied rewrites 72.8%

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

   if 2.6999999999999999e-5 < k < 1.7e155

   1. Initial program 35.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 37.6%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 75.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.8

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

   10. Applied rewrites 75.8%

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

   if 1.7e155 < k

   1. Initial program 35.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 37.6%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

   8. Applied rewrites 78.4%

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

Alternative 5: 88.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 2.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{{\left(\left|k\right|\right)}^{2} \cdot t}}{\left|k\right| \cdot \left|k\right|} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos \left(\left|k\right|\right)}{\left(t \cdot \left(\mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right) \cdot \left|k\right|\right)\right) \cdot \left|k\right|}\right)\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs k) 2.9e-5)
   (* (/ (* 2.0 (/ l (* (pow (fabs k) 2.0) t))) (* (fabs k) (fabs k))) l)
   (*
    2.0
    (*
     l
     (*
      l
      (/
       (cos (fabs k))
       (*
        (* t (* (fma (cos (+ (fabs k) (fabs k))) -0.5 0.5) (fabs k)))
        (fabs k))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 2.9e-5) {
		tmp = ((2.0 * (l / (pow(fabs(k), 2.0) * t))) / (fabs(k) * fabs(k))) * l;
	} else {
		tmp = 2.0 * (l * (l * (cos(fabs(k)) / ((t * (fma(cos((fabs(k) + fabs(k))), -0.5, 0.5) * fabs(k))) * fabs(k)))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 2.9e-5)
		tmp = Float64(Float64(Float64(2.0 * Float64(l / Float64((abs(k) ^ 2.0) * t))) / Float64(abs(k) * abs(k))) * l);
	else
		tmp = Float64(2.0 * Float64(l * Float64(l * Float64(cos(abs(k)) / Float64(Float64(t * Float64(fma(cos(Float64(abs(k) + abs(k))), -0.5, 0.5) * abs(k))) * abs(k))))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 2.9e-5], N[(N[(N[(2.0 * N[(l / N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(2.0 * N[(l * N[(l * N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] / N[(N[(t * N[(N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.9e-5

   1. Initial program 35.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 37.6%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 75.8%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 72.8

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

   11. Applied rewrites 72.8%

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

   if 2.9e-5 < k

   1. Initial program 35.5%

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

   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 73.3

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

   4. Applied rewrites 73.3%

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

      1. lift-/.f64 N/A

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

      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-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 82.1

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*r* N/A

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

   6. Applied rewrites 78.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. count-2 N/A

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

      9. sqr-sin-a-rev N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 85.3

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

   8. Applied rewrites 78.4%

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

Alternative 6: 88.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{{\left(\left|k\right|\right)}^{2} \cdot t}}{\left|k\right| \cdot \left|k\right|} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(\left|k\right|\right) \cdot \ell}{\left(\left(\mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right) \cdot t\right) \cdot \left|k\right|\right) \cdot \left|k\right|} \cdot \left(\ell + \ell\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs k) 2.7e-5)
   (* (/ (* 2.0 (/ l (* (pow (fabs k) 2.0) t))) (* (fabs k) (fabs k))) l)
   (*
    (/
     (* (cos (fabs k)) l)
     (*
      (* (* (fma (cos (+ (fabs k) (fabs k))) -0.5 0.5) t) (fabs k))
      (fabs k)))
    (+ l l))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 2.7e-5) {
		tmp = ((2.0 * (l / (pow(fabs(k), 2.0) * t))) / (fabs(k) * fabs(k))) * l;
	} else {
		tmp = ((cos(fabs(k)) * l) / (((fma(cos((fabs(k) + fabs(k))), -0.5, 0.5) * t) * fabs(k)) * fabs(k))) * (l + l);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 2.7e-5)
		tmp = Float64(Float64(Float64(2.0 * Float64(l / Float64((abs(k) ^ 2.0) * t))) / Float64(abs(k) * abs(k))) * l);
	else
		tmp = Float64(Float64(Float64(cos(abs(k)) * l) / Float64(Float64(Float64(fma(cos(Float64(abs(k) + abs(k))), -0.5, 0.5) * t) * abs(k)) * abs(k))) * Float64(l + l));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 2.7e-5], N[(N[(N[(2.0 * N[(l / N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] * l), $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] * N[(l + l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.6999999999999999e-5

   1. Initial program 35.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 37.6%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 75.8%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 72.8

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

   11. Applied rewrites 72.8%

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

   if 2.6999999999999999e-5 < k

   1. Initial program 35.5%

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

   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 73.3

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

   4. Applied rewrites 73.3%

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

      1. lift-/.f64 N/A

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

      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-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 82.1

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*r* N/A

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

   6. Applied rewrites 78.4%

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

   8. Applied rewrites 78.4%

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

Alternative 7: 75.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 2100000000:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{{\left(\left|k\right|\right)}^{2} \cdot t}}{\left|k\right| \cdot \left|k\right|} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{1 \cdot \ell}{\left(\mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right) \cdot t\right) \cdot \left|k\right|}}{\left|k\right|}\right) \cdot \ell\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs k) 2100000000.0)
   (* (/ (* 2.0 (/ l (* (pow (fabs k) 2.0) t))) (* (fabs k) (fabs k))) l)
   (*
    (*
     2.0
     (/
      (/
       (* 1.0 l)
       (* (* (fma (cos (+ (fabs k) (fabs k))) -0.5 0.5) t) (fabs k)))
      (fabs k)))
    l)))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 2100000000.0) {
		tmp = ((2.0 * (l / (pow(fabs(k), 2.0) * t))) / (fabs(k) * fabs(k))) * l;
	} else {
		tmp = (2.0 * (((1.0 * l) / ((fma(cos((fabs(k) + fabs(k))), -0.5, 0.5) * t) * fabs(k))) / fabs(k))) * l;
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 2100000000.0)
		tmp = Float64(Float64(Float64(2.0 * Float64(l / Float64((abs(k) ^ 2.0) * t))) / Float64(abs(k) * abs(k))) * l);
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(Float64(1.0 * l) / Float64(Float64(fma(cos(Float64(abs(k) + abs(k))), -0.5, 0.5) * t) * abs(k))) / abs(k))) * l);
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 2100000000.0], N[(N[(N[(2.0 * N[(l / N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(2.0 * N[(N[(N[(1.0 * l), $MachinePrecision] / N[(N[(N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.1e9

   1. Initial program 35.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 37.6%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 75.8%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 72.8

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

   11. Applied rewrites 72.8%

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

   if 2.1e9 < k

   1. Initial program 35.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 37.6%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. sqr-sin-a-rev N/A

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

      11. count-2 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

   8. Applied rewrites 82.3%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 65.4%

       \[\leadsto \left(2 \cdot \frac{\frac{1 \cdot \ell}{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right) \cdot k}}{k}\right) \cdot \ell \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 74.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 9.2 \cdot 10^{+254}:\\ \;\;\;\;\frac{2 \cdot \frac{\left|\ell\right|}{{k}^{2} \cdot t}}{k \cdot k} \cdot \left|\ell\right|\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left|\ell\right| \cdot \left(\left|\ell\right| \cdot \frac{\cos k}{\left(\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot k\right) \cdot k}\right)\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs l) 9.2e+254)
   (* (/ (* 2.0 (/ (fabs l) (* (pow k 2.0) t))) (* k k)) (fabs l))
   (*
    2.0
    (* (fabs l) (* (fabs l) (/ (cos k) (* (* (* (- 0.5 0.5) t) k) k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(l) <= 9.2e+254) {
		tmp = ((2.0 * (fabs(l) / (pow(k, 2.0) * t))) / (k * k)) * fabs(l);
	} else {
		tmp = 2.0 * (fabs(l) * (fabs(l) * (cos(k) / ((((0.5 - 0.5) * t) * k) * k))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (abs(l) <= 9.2d+254) then
        tmp = ((2.0d0 * (abs(l) / ((k ** 2.0d0) * t))) / (k * k)) * abs(l)
    else
        tmp = 2.0d0 * (abs(l) * (abs(l) * (cos(k) / ((((0.5d0 - 0.5d0) * t) * k) * k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(l) <= 9.2e+254) {
		tmp = ((2.0 * (Math.abs(l) / (Math.pow(k, 2.0) * t))) / (k * k)) * Math.abs(l);
	} else {
		tmp = 2.0 * (Math.abs(l) * (Math.abs(l) * (Math.cos(k) / ((((0.5 - 0.5) * t) * k) * k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(l) <= 9.2e+254:
		tmp = ((2.0 * (math.fabs(l) / (math.pow(k, 2.0) * t))) / (k * k)) * math.fabs(l)
	else:
		tmp = 2.0 * (math.fabs(l) * (math.fabs(l) * (math.cos(k) / ((((0.5 - 0.5) * t) * k) * k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(l) <= 9.2e+254)
		tmp = Float64(Float64(Float64(2.0 * Float64(abs(l) / Float64((k ^ 2.0) * t))) / Float64(k * k)) * abs(l));
	else
		tmp = Float64(2.0 * Float64(abs(l) * Float64(abs(l) * Float64(cos(k) / Float64(Float64(Float64(Float64(0.5 - 0.5) * t) * k) * k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(l) <= 9.2e+254)
		tmp = ((2.0 * (abs(l) / ((k ^ 2.0) * t))) / (k * k)) * abs(l);
	else
		tmp = 2.0 * (abs(l) * (abs(l) * (cos(k) / ((((0.5 - 0.5) * t) * k) * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[l], $MachinePrecision], 9.2e+254], N[(N[(N[(2.0 * N[(N[Abs[l], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Abs[l], $MachinePrecision] * N[(N[Abs[l], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(N[(0.5 - 0.5), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 9.19999999999999994e254

   1. Initial program 35.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 37.6%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 75.8%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 72.8

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

   11. Applied rewrites 72.8%

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

   if 9.19999999999999994e254 < l

   1. Initial program 35.5%

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

   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 73.3

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

   4. Applied rewrites 73.3%

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

      1. lift-/.f64 N/A

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

      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-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 82.1

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*r* N/A

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

   6. Applied rewrites 78.4%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 41.8%

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

Alternative 9: 72.8% accurate, 3.6× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ (* 2.0 (/ l (* (pow k 2.0) t))) (* k k)) l))

C

double code(double t, double l, double k) {
	return ((2.0 * (l / (pow(k, 2.0) * t))) / (k * 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 = ((2.0d0 * (l / ((k ** 2.0d0) * t))) / (k * k)) * l
end function

Java

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

Python

def code(t, l, k):
	return ((2.0 * (l / (math.pow(k, 2.0) * t))) / (k * k)) * l

Julia

function code(t, l, k)
	return Float64(Float64(Float64(2.0 * Float64(l / Float64((k ^ 2.0) * t))) / Float64(k * k)) * l)
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((2.0 * (l / ((k ^ 2.0) * t))) / (k * k)) * l;
end

Wolfram

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

Derivation

1. Initial program 35.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. associate-*l/ N/A

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

   12. associate-/r/ N/A

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

3. Applied rewrites 37.6%

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

4. Taylor expanded in t around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-sin.f64 82.3

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

6. Applied rewrites 82.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

8. Applied rewrites 75.8%

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

9. Taylor expanded in k around 0

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

    1. lower-*.f64 N/A

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

    2. lower-/.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-pow.f64 72.8

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

11. Applied rewrites 72.8%

    \[\leadsto \frac{2 \cdot \frac{\ell}{{k}^{2} \cdot t}}{\color{blue}{k} \cdot k} \cdot \ell \]
12. Add Preprocessing

Alternative 10: 70.7% accurate, 3.9× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* (* 2.0 (/ (/ l (* (pow k 3.0) t)) k)) l))

C

double code(double t, double l, double k) {
	return (2.0 * ((l / (pow(k, 3.0) * 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 = (2.0d0 * ((l / ((k ** 3.0d0) * t)) / k)) * l
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. associate-*l/ N/A

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

   12. associate-/r/ N/A

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

3. Applied rewrites 37.6%

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

4. Taylor expanded in t around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-sin.f64 82.3

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

6. Applied rewrites 82.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. sqr-sin-a-rev N/A

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

   11. count-2 N/A

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

   12. lift-+.f64 N/A

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

   13. lift-cos.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift--.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. associate-*r* N/A

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

   19. *-commutative N/A

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

   20. lift-*.f64 N/A

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

8. Applied rewrites 82.3%

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

9. Taylor expanded in k around 0

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

    1. lower-/.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lower-pow.f64 70.7

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

11. Applied rewrites 70.7%

    \[\leadsto \left(2 \cdot \frac{\frac{\ell}{{k}^{3} \cdot t}}{k}\right) \cdot \ell \]
12. Add Preprocessing

Alternative 11: 69.5% accurate, 4.4× speedup

Math

\[\frac{{k}^{-4} \cdot \ell}{t} \cdot \left(\ell + \ell\right) \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ (* (pow k -4.0) l) t) (+ l l)))

C

double code(double t, double l, double k) {
	return ((pow(k, -4.0) * l) / t) * (l + 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 = (((k ** (-4.0d0)) * l) / t) * (l + l)
end function

Java

public static double code(double t, double l, double k) {
	return ((Math.pow(k, -4.0) * l) / t) * (l + l);
}

Python

def code(t, l, k):
	return ((math.pow(k, -4.0) * l) / t) * (l + l)

Julia

function code(t, l, k)
	return Float64(Float64(Float64((k ^ -4.0) * l) / t) * Float64(l + l))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (((k ^ -4.0) * l) / t) * (l + l);
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(N[Power[k, -4.0], $MachinePrecision] * l), $MachinePrecision] / t), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.5%

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

2. Taylor expanded in k around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 62.5

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

4. Applied rewrites 62.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. pow2 N/A

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{{k}^{4}} \cdot t} \]

   5. lift-*.f64 N/A

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{{k}^{4}} \cdot t} \]

   6. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{k}^{4} \cdot t}} \]

   7. count-2-rev N/A

      \[\leadsto \frac{\ell \cdot \ell + \ell \cdot \ell}{\color{blue}{{k}^{4}} \cdot t} \]

   8. div-add-rev N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} + \color{blue}{\frac{\ell \cdot \ell}{{k}^{4} \cdot t}} \]

   9. mult-flip N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{1}{{k}^{4} \cdot t} + \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   10. mult-flip N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{1}{{k}^{4} \cdot t} + \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}} \]

   11. distribute-rgt-out N/A

       \[\leadsto \frac{1}{{k}^{4} \cdot t} \cdot \color{blue}{\left(\ell \cdot \ell + \ell \cdot \ell\right)} \]

   12. count-2-rev N/A

       \[\leadsto \frac{1}{{k}^{4} \cdot t} \cdot \left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{1}{{k}^{4} \cdot t} \cdot \color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right)} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{1}{{k}^{4} \cdot t} \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right) \]

   15. associate-/r* N/A

       \[\leadsto \frac{\frac{1}{{k}^{4}}}{t} \cdot \left(\color{blue}{2} \cdot \left(\ell \cdot \ell\right)\right) \]

   16. lower-/.f64 N/A

       \[\leadsto \frac{\frac{1}{{k}^{4}}}{t} \cdot \left(\color{blue}{2} \cdot \left(\ell \cdot \ell\right)\right) \]

   17. lift-pow.f64 N/A

       \[\leadsto \frac{\frac{1}{{k}^{4}}}{t} \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right) \]

   18. pow-flip N/A

       \[\leadsto \frac{{k}^{\left(\mathsf{neg}\left(4\right)\right)}}{t} \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right) \]

   19. lower-pow.f64 N/A

       \[\leadsto \frac{{k}^{\left(\mathsf{neg}\left(4\right)\right)}}{t} \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right) \]

   20. metadata-eval N/A

       \[\leadsto \frac{{k}^{-4}}{t} \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right) \]

   21. *-commutative N/A

       \[\leadsto \frac{{k}^{-4}}{t} \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{2}\right) \]

   22. lower-*.f64 62.4

       \[\leadsto \frac{{k}^{-4}}{t} \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{2}\right) \]

6. Applied rewrites 62.4%

   \[\leadsto \frac{{k}^{-4}}{t} \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot 2\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{{k}^{-4}}{t} \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot 2\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{{k}^{-4}}{t} \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{2}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{{k}^{-4}}{t} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right) \]

   4. associate-*l* N/A

      \[\leadsto \frac{{k}^{-4}}{t} \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot 2\right)}\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{{k}^{-4}}{t} \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{2}\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \left(\frac{{k}^{-4}}{t} \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot 2\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \left(\frac{{k}^{-4}}{t} \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot 2\right)} \]

   8. lower-*.f64 68.7

      \[\leadsto \left(\frac{{k}^{-4}}{t} \cdot \ell\right) \cdot \left(\color{blue}{\ell} \cdot 2\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \left(\frac{{k}^{-4}}{t} \cdot \ell\right) \cdot \left(\ell \cdot \color{blue}{2}\right) \]

   10. *-commutative N/A

       \[\leadsto \left(\frac{{k}^{-4}}{t} \cdot \ell\right) \cdot \left(2 \cdot \color{blue}{\ell}\right) \]

   11. count-2-rev N/A

       \[\leadsto \left(\frac{{k}^{-4}}{t} \cdot \ell\right) \cdot \left(\ell + \color{blue}{\ell}\right) \]

   12. lower-+.f64 68.7

       \[\leadsto \left(\frac{{k}^{-4}}{t} \cdot \ell\right) \cdot \left(\ell + \color{blue}{\ell}\right) \]

8. Applied rewrites 68.7%

   \[\leadsto \color{blue}{\left(\frac{{k}^{-4}}{t} \cdot \ell\right) \cdot \left(\ell + \ell\right)} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\frac{{k}^{-4}}{t} \cdot \ell\right) \cdot \left(\color{blue}{\ell} + \ell\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \left(\frac{{k}^{-4}}{t} \cdot \ell\right) \cdot \left(\ell + \ell\right) \]

   3. associate-*l/ N/A

      \[\leadsto \frac{{k}^{-4} \cdot \ell}{t} \cdot \left(\color{blue}{\ell} + \ell\right) \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{{k}^{-4} \cdot \ell}{t} \cdot \left(\color{blue}{\ell} + \ell\right) \]

   5. lower-*.f64 69.5

      \[\leadsto \frac{{k}^{-4} \cdot \ell}{t} \cdot \left(\ell + \ell\right) \]

10. Applied rewrites 69.5%

    \[\leadsto \frac{{k}^{-4} \cdot \ell}{t} \cdot \left(\color{blue}{\ell} + \ell\right) \]
11. Add Preprocessing

Alternative 12: 68.9% accurate, 4.4× speedup

Math

\[\frac{\ell}{{k}^{4} \cdot t} \cdot \left(\ell + \ell\right) \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ l (* (pow k 4.0) t)) (+ l l)))

C

double code(double t, double l, double k) {
	return (l / (pow(k, 4.0) * t)) * (l + 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 ** 4.0d0) * t)) * (l + l)
end function

Java

public static double code(double t, double l, double k) {
	return (l / (Math.pow(k, 4.0) * t)) * (l + l);
}

Python

def code(t, l, k):
	return (l / (math.pow(k, 4.0) * t)) * (l + l)

Julia

function code(t, l, k)
	return Float64(Float64(l / Float64((k ^ 4.0) * t)) * Float64(l + l))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / ((k ^ 4.0) * t)) * (l + l);
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[(N[Power[k, 4.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

   5. lower-pow.f64 62.5

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

4. Applied rewrites 62.5%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lift-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lift-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. pow2 N/A

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{{k}^{4}} \cdot t} \]

   5. lift-*.f64 N/A

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{{k}^{4}} \cdot t} \]

   6. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{k}^{4} \cdot t}} \]

   7. count-2-rev N/A

      \[\leadsto \frac{\ell \cdot \ell + \ell \cdot \ell}{\color{blue}{{k}^{4}} \cdot t} \]

   8. div-add-rev N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} + \color{blue}{\frac{\ell \cdot \ell}{{k}^{4} \cdot t}} \]

   9. mult-flip N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{1}{{k}^{4} \cdot t} + \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   10. mult-flip N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{1}{{k}^{4} \cdot t} + \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{1}{{k}^{4} \cdot t}} \]

   11. distribute-rgt-out N/A

       \[\leadsto \frac{1}{{k}^{4} \cdot t} \cdot \color{blue}{\left(\ell \cdot \ell + \ell \cdot \ell\right)} \]

   12. count-2-rev N/A

       \[\leadsto \frac{1}{{k}^{4} \cdot t} \cdot \left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{1}{{k}^{4} \cdot t} \cdot \color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right)} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{1}{{k}^{4} \cdot t} \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right) \]

   15. associate-/r* N/A

       \[\leadsto \frac{\frac{1}{{k}^{4}}}{t} \cdot \left(\color{blue}{2} \cdot \left(\ell \cdot \ell\right)\right) \]

   16. lower-/.f64 N/A

       \[\leadsto \frac{\frac{1}{{k}^{4}}}{t} \cdot \left(\color{blue}{2} \cdot \left(\ell \cdot \ell\right)\right) \]

   17. lift-pow.f64 N/A

       \[\leadsto \frac{\frac{1}{{k}^{4}}}{t} \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right) \]

   18. pow-flip N/A

       \[\leadsto \frac{{k}^{\left(\mathsf{neg}\left(4\right)\right)}}{t} \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right) \]

   19. lower-pow.f64 N/A

       \[\leadsto \frac{{k}^{\left(\mathsf{neg}\left(4\right)\right)}}{t} \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right) \]

   20. metadata-eval N/A

       \[\leadsto \frac{{k}^{-4}}{t} \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right) \]

   21. *-commutative N/A

       \[\leadsto \frac{{k}^{-4}}{t} \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{2}\right) \]

   22. lower-*.f64 62.4

       \[\leadsto \frac{{k}^{-4}}{t} \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{2}\right) \]

6. Applied rewrites 62.4%

   \[\leadsto \frac{{k}^{-4}}{t} \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot 2\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{{k}^{-4}}{t} \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot 2\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{{k}^{-4}}{t} \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{2}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{{k}^{-4}}{t} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right) \]

   4. associate-*l* N/A

      \[\leadsto \frac{{k}^{-4}}{t} \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot 2\right)}\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{{k}^{-4}}{t} \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{2}\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \left(\frac{{k}^{-4}}{t} \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot 2\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \left(\frac{{k}^{-4}}{t} \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot 2\right)} \]

   8. lower-*.f64 68.7

      \[\leadsto \left(\frac{{k}^{-4}}{t} \cdot \ell\right) \cdot \left(\color{blue}{\ell} \cdot 2\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \left(\frac{{k}^{-4}}{t} \cdot \ell\right) \cdot \left(\ell \cdot \color{blue}{2}\right) \]

   10. *-commutative N/A

       \[\leadsto \left(\frac{{k}^{-4}}{t} \cdot \ell\right) \cdot \left(2 \cdot \color{blue}{\ell}\right) \]

   11. count-2-rev N/A

       \[\leadsto \left(\frac{{k}^{-4}}{t} \cdot \ell\right) \cdot \left(\ell + \color{blue}{\ell}\right) \]

   12. lower-+.f64 68.7

       \[\leadsto \left(\frac{{k}^{-4}}{t} \cdot \ell\right) \cdot \left(\ell + \color{blue}{\ell}\right) \]

8. Applied rewrites 68.7%

   \[\leadsto \color{blue}{\left(\frac{{k}^{-4}}{t} \cdot \ell\right) \cdot \left(\ell + \ell\right)} \]

9. Taylor expanded in t around 0

   \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \left(\color{blue}{\ell} + \ell\right) \]
10. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \left(\ell + \ell\right) \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \left(\ell + \ell\right) \]

    3. lower-pow.f64 68.9

       \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \left(\ell + \ell\right) \]

11. Applied rewrites 68.9%

    \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \left(\color{blue}{\ell} + \ell\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025176 
(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))))