Toniolo and Linder, Equation (10-)

Percentage Accurate: 37.2% → 90.3%

Time: 8.8s

Alternatives: 13

Speedup: 4.4×

• 60.1% 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: 37.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 2.65 \cdot 10^{-120}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\left|\ell\right|}{t}, \frac{\left|\ell\right|}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \left|\ell\right|\right) \cdot 2\\ \mathbf{elif}\;\left|\ell\right| \leq 3.2 \cdot 10^{+36}:\\ \;\;\;\;2 \cdot \left(\left|\ell\right| \cdot \left(\left|\ell\right| \cdot \frac{\cos k}{\left(\left(\frac{1}{{\sin k}^{-2}} \cdot t\right) \cdot k\right) \cdot k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left|\ell\right| + \left|\ell\right|}{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right) \cdot k} \cdot \frac{\cos k}{k}\right) \cdot \left|\ell\right|\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs l) 2.65e-120)
   (*
    (*
     (/
      (fma -0.16666666666666666 (/ (fabs l) t) (/ (fabs l) (* (pow k 2.0) t)))
      (pow k 2.0))
     (fabs l))
    2.0)
   (if (<= (fabs l) 3.2e+36)
     (*
      2.0
      (*
       (fabs l)
       (* (fabs l) (/ (cos k) (* (* (* (/ 1.0 (pow (sin k) -2.0)) t) k) k)))))
     (*
      (*
       (/ (+ (fabs l) (fabs l)) (* (* (fma (cos (+ k k)) -0.5 0.5) t) k))
       (/ (cos k) k))
      (fabs l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(l) <= 2.65e-120) {
		tmp = ((fma(-0.16666666666666666, (fabs(l) / t), (fabs(l) / (pow(k, 2.0) * t))) / pow(k, 2.0)) * fabs(l)) * 2.0;
	} else if (fabs(l) <= 3.2e+36) {
		tmp = 2.0 * (fabs(l) * (fabs(l) * (cos(k) / ((((1.0 / pow(sin(k), -2.0)) * t) * k) * k))));
	} else {
		tmp = (((fabs(l) + fabs(l)) / ((fma(cos((k + k)), -0.5, 0.5) * t) * k)) * (cos(k) / k)) * fabs(l);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(l) <= 2.65e-120)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(abs(l) / t), Float64(abs(l) / Float64((k ^ 2.0) * t))) / (k ^ 2.0)) * abs(l)) * 2.0);
	elseif (abs(l) <= 3.2e+36)
		tmp = Float64(2.0 * Float64(abs(l) * Float64(abs(l) * Float64(cos(k) / Float64(Float64(Float64(Float64(1.0 / (sin(k) ^ -2.0)) * t) * k) * k)))));
	else
		tmp = Float64(Float64(Float64(Float64(abs(l) + abs(l)) / Float64(Float64(fma(cos(Float64(k + k)), -0.5, 0.5) * t) * k)) * Float64(cos(k) / k)) * abs(l));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[l], $MachinePrecision], 2.65e-120], N[(N[(N[(N[(-0.16666666666666666 * N[(N[Abs[l], $MachinePrecision] / t), $MachinePrecision] + N[(N[Abs[l], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[N[Abs[l], $MachinePrecision], 3.2e+36], N[(2.0 * N[(N[Abs[l], $MachinePrecision] * N[(N[Abs[l], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(N[(1.0 / N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Abs[l], $MachinePrecision] + N[Abs[l], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.64999999999999999e-120

   1. Initial program 37.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 31.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 38.5%

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

   6. Taylor expanded in k around 0

      \[\leadsto \left(\color{blue}{\frac{\frac{-1}{6} \cdot \frac{{k}^{2} \cdot \ell}{t} + \frac{\ell}{t}}{{k}^{4}}} \cdot \ell\right) \cdot 2 \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{\frac{-1}{6} \cdot \frac{{k}^{2} \cdot \ell}{t} + \frac{\ell}{t}}{\color{blue}{{k}^{4}}} \cdot \ell\right) \cdot 2 \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{\color{blue}{k}}^{4}} \cdot \ell\right) \cdot 2 \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      7. lower-pow.f64 51.1%

         \[\leadsto \left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{\color{blue}{4}}} \cdot \ell\right) \cdot 2 \]

   8. Applied rewrites 51.1%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}}} \cdot \ell\right) \cdot 2 \]

   9. Taylor expanded in k around inf

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

       1. lower-/.f64 N/A

          \[\leadsto \left(\frac{\frac{-1}{6} \cdot \frac{\ell}{t} + \frac{\ell}{{k}^{2} \cdot t}}{{k}^{\color{blue}{2}}} \cdot \ell\right) \cdot 2 \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       3. lower-/.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       4. lower-/.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       6. lower-pow.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       7. lower-pow.f64 67.4%

          \[\leadsto \left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

   11. Applied rewrites 67.4%

       \[\leadsto \left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{\color{blue}{{k}^{2}}} \cdot \ell\right) \cdot 2 \]

   if 2.64999999999999999e-120 < l < 3.1999999999999999e36

   1. Initial program 37.2%

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

   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 74.5%

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

   4. Applied rewrites 74.5%

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

      1. lift-/.f64 N/A

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

      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 82.8%

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

      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 79.2%

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}\right)}\right) \]
   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. lift-cos.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. count-2 N/A

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

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

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

      7. pow2 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. pow-neg N/A

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

      11. lower-unsound-/.f64 N/A

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

      12. lower-unsound-pow.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. metadata-eval 86.5%

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

   8. Applied rewrites 86.5%

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

   if 3.1999999999999999e36 < l

   1. Initial program 37.2%

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

   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 74.5%

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

   4. Applied rewrites 74.5%

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

      1. lift-/.f64 N/A

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

      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 82.8%

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

      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 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   8. Applied rewrites 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 82.9%

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

   10. Applied rewrites 82.9%

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

Alternative 2: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \cos \left(\left|k\right|\right)\\ \mathbf{if}\;\left|k\right| \leq 2 \cdot 10^{+36}:\\ \;\;\;\;\left(\frac{\ell \cdot t\_1}{{\left(\left|k\right|\right)}^{2} \cdot \left(t \cdot {\sin \left(\left|k\right|\right)}^{2}\right)} \cdot \ell\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell + \ell}{\left(\mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right) \cdot t\right) \cdot \left|k\right|} \cdot \frac{t\_1}{\left|k\right|}\right) \cdot \ell\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cos (fabs k))))
   (if (<= (fabs k) 2e+36)
     (*
      (* (/ (* l t_1) (* (pow (fabs k) 2.0) (* t (pow (sin (fabs k)) 2.0)))) l)
      2.0)
     (*
      (*
       (/
        (+ l l)
        (* (* (fma (cos (+ (fabs k) (fabs k))) -0.5 0.5) t) (fabs k)))
       (/ t_1 (fabs k)))
      l))))

C

double code(double t, double l, double k) {
	double t_1 = cos(fabs(k));
	double tmp;
	if (fabs(k) <= 2e+36) {
		tmp = (((l * t_1) / (pow(fabs(k), 2.0) * (t * pow(sin(fabs(k)), 2.0)))) * l) * 2.0;
	} else {
		tmp = (((l + l) / ((fma(cos((fabs(k) + fabs(k))), -0.5, 0.5) * t) * fabs(k))) * (t_1 / fabs(k))) * l;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cos(abs(k))
	tmp = 0.0
	if (abs(k) <= 2e+36)
		tmp = Float64(Float64(Float64(Float64(l * t_1) / Float64((abs(k) ^ 2.0) * Float64(t * (sin(abs(k)) ^ 2.0)))) * l) * 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(l + l) / Float64(Float64(fma(cos(Float64(abs(k) + abs(k))), -0.5, 0.5) * t) * abs(k))) * Float64(t_1 / 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], 2e+36], N[(N[(N[(N[(l * t$95$1), $MachinePrecision] / N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(l + 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[(t$95$1 / N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.00000000000000008e36

   1. Initial program 37.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 31.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 38.5%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 83.0%

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

   8. Applied rewrites 83.0%

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

   if 2.00000000000000008e36 < k

   1. Initial program 37.2%

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

   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 74.5%

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

   4. Applied rewrites 74.5%

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

      1. lift-/.f64 N/A

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

      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 82.8%

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

      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 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   8. Applied rewrites 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 82.9%

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

   10. Applied rewrites 82.9%

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

Alternative 3: 89.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \cos \left(\left|k\right|\right)\\ \mathbf{if}\;\left|k\right| \leq 3.5 \cdot 10^{-10}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{t\_1}{\left(\left({\left(\left|k\right|\right)}^{2} \cdot t\right) \cdot \left|k\right|\right) \cdot \left|k\right|}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell + \ell}{\left(\mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right) \cdot t\right) \cdot \left|k\right|} \cdot \frac{t\_1}{\left|k\right|}\right) \cdot \ell\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cos (fabs k))))
   (if (<= (fabs k) 3.5e-10)
     (*
      2.0
      (* l (* l (/ t_1 (* (* (* (pow (fabs k) 2.0) t) (fabs k)) (fabs k))))))
     (*
      (*
       (/
        (+ l l)
        (* (* (fma (cos (+ (fabs k) (fabs k))) -0.5 0.5) t) (fabs k)))
       (/ t_1 (fabs k)))
      l))))

C

double code(double t, double l, double k) {
	double t_1 = cos(fabs(k));
	double tmp;
	if (fabs(k) <= 3.5e-10) {
		tmp = 2.0 * (l * (l * (t_1 / (((pow(fabs(k), 2.0) * t) * fabs(k)) * fabs(k)))));
	} else {
		tmp = (((l + l) / ((fma(cos((fabs(k) + fabs(k))), -0.5, 0.5) * t) * fabs(k))) * (t_1 / fabs(k))) * l;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cos(abs(k))
	tmp = 0.0
	if (abs(k) <= 3.5e-10)
		tmp = Float64(2.0 * Float64(l * Float64(l * Float64(t_1 / Float64(Float64(Float64((abs(k) ^ 2.0) * t) * abs(k)) * abs(k))))));
	else
		tmp = Float64(Float64(Float64(Float64(l + l) / Float64(Float64(fma(cos(Float64(abs(k) + abs(k))), -0.5, 0.5) * t) * abs(k))) * Float64(t_1 / 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], 3.5e-10], N[(2.0 * N[(l * N[(l * N[(t$95$1 / N[(N[(N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l + 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[(t$95$1 / N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.4999999999999998e-10

   1. Initial program 37.2%

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

   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 74.5%

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

   4. Applied rewrites 74.5%

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

      1. lift-/.f64 N/A

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

      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 82.8%

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

      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 79.2%

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

   7. Taylor expanded in k around 0

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

      1. lower-pow.f64 71.9%

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

   9. Applied rewrites 71.9%

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

   if 3.4999999999999998e-10 < k

   1. Initial program 37.2%

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

   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 74.5%

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

   4. Applied rewrites 74.5%

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

      1. lift-/.f64 N/A

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

      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 82.8%

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

      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 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   8. Applied rewrites 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 82.9%

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

   10. Applied rewrites 82.9%

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

Alternative 4: 85.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \cos \left(\left|k\right|\right)\\ \mathbf{if}\;\left|k\right| \leq 3.5 \cdot 10^{-10}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{t\_1}{\left(\left({\left(\left|k\right|\right)}^{2} \cdot t\right) \cdot \left|k\right|\right) \cdot \left|k\right|}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 \cdot \ell}{\left(\left(\mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right) \cdot t\right) \cdot \left|k\right|\right) \cdot \left|k\right|} \cdot \left(\ell + \ell\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cos (fabs k))))
   (if (<= (fabs k) 3.5e-10)
     (*
      2.0
      (* l (* l (/ t_1 (* (* (* (pow (fabs k) 2.0) t) (fabs k)) (fabs k))))))
     (*
      (/
       (* t_1 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 t_1 = cos(fabs(k));
	double tmp;
	if (fabs(k) <= 3.5e-10) {
		tmp = 2.0 * (l * (l * (t_1 / (((pow(fabs(k), 2.0) * t) * fabs(k)) * fabs(k)))));
	} else {
		tmp = ((t_1 * 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)
	t_1 = cos(abs(k))
	tmp = 0.0
	if (abs(k) <= 3.5e-10)
		tmp = Float64(2.0 * Float64(l * Float64(l * Float64(t_1 / Float64(Float64(Float64((abs(k) ^ 2.0) * t) * abs(k)) * abs(k))))));
	else
		tmp = Float64(Float64(Float64(t_1 * 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_] := Block[{t$95$1 = N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 3.5e-10], N[(2.0 * N[(l * N[(l * N[(t$95$1 / N[(N[(N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 * 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 < 3.4999999999999998e-10

   1. Initial program 37.2%

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

   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 74.5%

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

   4. Applied rewrites 74.5%

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

      1. lift-/.f64 N/A

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

      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 82.8%

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

      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 79.2%

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

   7. Taylor expanded in k around 0

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

      1. lower-pow.f64 71.9%

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

   9. Applied rewrites 71.9%

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

   if 3.4999999999999998e-10 < k

   1. Initial program 37.2%

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

   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 74.5%

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

   4. Applied rewrites 74.5%

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

      1. lift-/.f64 N/A

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

      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 82.8%

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

      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 79.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   8. Applied rewrites 79.2%

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

Alternative 5: 75.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 5 \cdot 10^{-118}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\left|\ell\right|}{t}, \frac{\left|\ell\right|}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \left|\ell\right|\right) \cdot 2\\ \mathbf{elif}\;\left|\ell\right| \leq 1.55 \cdot 10^{+186}:\\ \;\;\;\;2 \cdot \left(\left|\ell\right| \cdot \left(\left|\ell\right| \cdot \frac{1}{\left(\left(\frac{1}{{\sin k}^{-2}} \cdot t\right) \cdot k\right) \cdot k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{\left(\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \left(\left(\left|\ell\right| \cdot \left|\ell\right|\right) \cdot 2\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs l) 5e-118)
   (*
    (*
     (/
      (fma -0.16666666666666666 (/ (fabs l) t) (/ (fabs l) (* (pow k 2.0) t)))
      (pow k 2.0))
     (fabs l))
    2.0)
   (if (<= (fabs l) 1.55e+186)
     (*
      2.0
      (*
       (fabs l)
       (* (fabs l) (/ 1.0 (* (* (* (/ 1.0 (pow (sin k) -2.0)) t) k) k)))))
     (*
      (/ (cos k) (* (* (* (- 0.5 0.5) t) k) k))
      (* (* (fabs l) (fabs l)) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(l) <= 5e-118) {
		tmp = ((fma(-0.16666666666666666, (fabs(l) / t), (fabs(l) / (pow(k, 2.0) * t))) / pow(k, 2.0)) * fabs(l)) * 2.0;
	} else if (fabs(l) <= 1.55e+186) {
		tmp = 2.0 * (fabs(l) * (fabs(l) * (1.0 / ((((1.0 / pow(sin(k), -2.0)) * t) * k) * k))));
	} else {
		tmp = (cos(k) / ((((0.5 - 0.5) * t) * k) * k)) * ((fabs(l) * fabs(l)) * 2.0);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(l) <= 5e-118)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(abs(l) / t), Float64(abs(l) / Float64((k ^ 2.0) * t))) / (k ^ 2.0)) * abs(l)) * 2.0);
	elseif (abs(l) <= 1.55e+186)
		tmp = Float64(2.0 * Float64(abs(l) * Float64(abs(l) * Float64(1.0 / Float64(Float64(Float64(Float64(1.0 / (sin(k) ^ -2.0)) * t) * k) * k)))));
	else
		tmp = Float64(Float64(cos(k) / Float64(Float64(Float64(Float64(0.5 - 0.5) * t) * k) * k)) * Float64(Float64(abs(l) * abs(l)) * 2.0));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[l], $MachinePrecision], 5e-118], N[(N[(N[(N[(-0.16666666666666666 * N[(N[Abs[l], $MachinePrecision] / t), $MachinePrecision] + N[(N[Abs[l], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[N[Abs[l], $MachinePrecision], 1.55e+186], N[(2.0 * N[(N[Abs[l], $MachinePrecision] * N[(N[Abs[l], $MachinePrecision] * N[(1.0 / N[(N[(N[(N[(1.0 / N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(N[(0.5 - 0.5), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 5.00000000000000015e-118

   1. Initial program 37.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 31.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 38.5%

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

   6. Taylor expanded in k around 0

      \[\leadsto \left(\color{blue}{\frac{\frac{-1}{6} \cdot \frac{{k}^{2} \cdot \ell}{t} + \frac{\ell}{t}}{{k}^{4}}} \cdot \ell\right) \cdot 2 \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{\frac{-1}{6} \cdot \frac{{k}^{2} \cdot \ell}{t} + \frac{\ell}{t}}{\color{blue}{{k}^{4}}} \cdot \ell\right) \cdot 2 \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{\color{blue}{k}}^{4}} \cdot \ell\right) \cdot 2 \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      7. lower-pow.f64 51.1%

         \[\leadsto \left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{\color{blue}{4}}} \cdot \ell\right) \cdot 2 \]

   8. Applied rewrites 51.1%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}}} \cdot \ell\right) \cdot 2 \]

   9. Taylor expanded in k around inf

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

       1. lower-/.f64 N/A

          \[\leadsto \left(\frac{\frac{-1}{6} \cdot \frac{\ell}{t} + \frac{\ell}{{k}^{2} \cdot t}}{{k}^{\color{blue}{2}}} \cdot \ell\right) \cdot 2 \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       3. lower-/.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       4. lower-/.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       6. lower-pow.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       7. lower-pow.f64 67.4%

          \[\leadsto \left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

   11. Applied rewrites 67.4%

       \[\leadsto \left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{\color{blue}{{k}^{2}}} \cdot \ell\right) \cdot 2 \]

   if 5.00000000000000015e-118 < l < 1.5500000000000001e186

   1. Initial program 37.2%

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

   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 74.5%

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

   4. Applied rewrites 74.5%

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

      1. lift-/.f64 N/A

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

      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 82.8%

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

      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 79.2%

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}\right)}\right) \]
   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. lift-cos.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. count-2 N/A

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

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

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

      7. pow2 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. pow-neg N/A

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

      11. lower-unsound-/.f64 N/A

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

      12. lower-unsound-pow.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. metadata-eval 86.5%

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

   8. Applied rewrites 86.5%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 72.8%

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

   if 1.5500000000000001e186 < l

   1. Initial program 37.2%

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

   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 74.5%

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

   4. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

   6. Applied rewrites 70.9%

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

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\cos k}{\left(\left(\left(0.5 - \frac{1}{2}\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 37.1%

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

Alternative 6: 75.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 2.2 \cdot 10^{+17}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\left|\ell\right|}{t}, \frac{\left|\ell\right|}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \left|\ell\right|\right) \cdot 2\\ \mathbf{elif}\;\left|\ell\right| \leq 1.7 \cdot 10^{+186}:\\ \;\;\;\;2 \cdot \left(\left|\ell\right| \cdot \left(\left|\ell\right| \cdot \frac{\cos k}{\left({k}^{3} \cdot t\right) \cdot k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{\left(\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \left(\left(\left|\ell\right| \cdot \left|\ell\right|\right) \cdot 2\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs l) 2.2e+17)
   (*
    (*
     (/
      (fma -0.16666666666666666 (/ (fabs l) t) (/ (fabs l) (* (pow k 2.0) t)))
      (pow k 2.0))
     (fabs l))
    2.0)
   (if (<= (fabs l) 1.7e+186)
     (* 2.0 (* (fabs l) (* (fabs l) (/ (cos k) (* (* (pow k 3.0) t) k)))))
     (*
      (/ (cos k) (* (* (* (- 0.5 0.5) t) k) k))
      (* (* (fabs l) (fabs l)) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(l) <= 2.2e+17) {
		tmp = ((fma(-0.16666666666666666, (fabs(l) / t), (fabs(l) / (pow(k, 2.0) * t))) / pow(k, 2.0)) * fabs(l)) * 2.0;
	} else if (fabs(l) <= 1.7e+186) {
		tmp = 2.0 * (fabs(l) * (fabs(l) * (cos(k) / ((pow(k, 3.0) * t) * k))));
	} else {
		tmp = (cos(k) / ((((0.5 - 0.5) * t) * k) * k)) * ((fabs(l) * fabs(l)) * 2.0);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(l) <= 2.2e+17)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(abs(l) / t), Float64(abs(l) / Float64((k ^ 2.0) * t))) / (k ^ 2.0)) * abs(l)) * 2.0);
	elseif (abs(l) <= 1.7e+186)
		tmp = Float64(2.0 * Float64(abs(l) * Float64(abs(l) * Float64(cos(k) / Float64(Float64((k ^ 3.0) * t) * k)))));
	else
		tmp = Float64(Float64(cos(k) / Float64(Float64(Float64(Float64(0.5 - 0.5) * t) * k) * k)) * Float64(Float64(abs(l) * abs(l)) * 2.0));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[l], $MachinePrecision], 2.2e+17], N[(N[(N[(N[(-0.16666666666666666 * N[(N[Abs[l], $MachinePrecision] / t), $MachinePrecision] + N[(N[Abs[l], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[N[Abs[l], $MachinePrecision], 1.7e+186], N[(2.0 * N[(N[Abs[l], $MachinePrecision] * N[(N[Abs[l], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[(N[Power[k, 3.0], $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(N[(0.5 - 0.5), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.2e17

   1. Initial program 37.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 31.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 38.5%

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

   6. Taylor expanded in k around 0

      \[\leadsto \left(\color{blue}{\frac{\frac{-1}{6} \cdot \frac{{k}^{2} \cdot \ell}{t} + \frac{\ell}{t}}{{k}^{4}}} \cdot \ell\right) \cdot 2 \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{\frac{-1}{6} \cdot \frac{{k}^{2} \cdot \ell}{t} + \frac{\ell}{t}}{\color{blue}{{k}^{4}}} \cdot \ell\right) \cdot 2 \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{\color{blue}{k}}^{4}} \cdot \ell\right) \cdot 2 \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      7. lower-pow.f64 51.1%

         \[\leadsto \left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{\color{blue}{4}}} \cdot \ell\right) \cdot 2 \]

   8. Applied rewrites 51.1%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}}} \cdot \ell\right) \cdot 2 \]

   9. Taylor expanded in k around inf

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

       1. lower-/.f64 N/A

          \[\leadsto \left(\frac{\frac{-1}{6} \cdot \frac{\ell}{t} + \frac{\ell}{{k}^{2} \cdot t}}{{k}^{\color{blue}{2}}} \cdot \ell\right) \cdot 2 \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       3. lower-/.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       4. lower-/.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       6. lower-pow.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       7. lower-pow.f64 67.4%

          \[\leadsto \left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

   11. Applied rewrites 67.4%

       \[\leadsto \left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{\color{blue}{{k}^{2}}} \cdot \ell\right) \cdot 2 \]

   if 2.2e17 < l < 1.70000000000000003e186

   1. Initial program 37.2%

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

   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 74.5%

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

   4. Applied rewrites 74.5%

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

      1. lift-/.f64 N/A

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

      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 82.8%

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

      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 79.2%

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

   7. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 71.2%

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

   9. Applied rewrites 71.2%

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

   if 1.70000000000000003e186 < l

   1. Initial program 37.2%

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

   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 74.5%

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

   4. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

   6. Applied rewrites 70.9%

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

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\cos k}{\left(\left(\left(0.5 - \frac{1}{2}\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 37.1%

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

Alternative 7: 74.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 2 \cdot 10^{+40}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\left|\ell\right|}{t}, \frac{\left|\ell\right|}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \left|\ell\right|\right) \cdot 2\\ \mathbf{elif}\;\left|\ell\right| \leq 1.65 \cdot 10^{+186}:\\ \;\;\;\;\left(\left|\ell\right| + \left|\ell\right|\right) \cdot \frac{\left|\ell\right|}{\mathsf{fma}\left(t \cdot \mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, t\right) \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{\left(\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \left(\left(\left|\ell\right| \cdot \left|\ell\right|\right) \cdot 2\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs l) 2e+40)
   (*
    (*
     (/
      (fma -0.16666666666666666 (/ (fabs l) t) (/ (fabs l) (* (pow k 2.0) t)))
      (pow k 2.0))
     (fabs l))
    2.0)
   (if (<= (fabs l) 1.65e+186)
     (*
      (+ (fabs l) (fabs l))
      (/
       (fabs l)
       (*
        (fma
         (* t (fma 0.08611111111111111 (* k k) 0.16666666666666666))
         (* k k)
         t)
        (pow k 4.0))))
     (*
      (/ (cos k) (* (* (* (- 0.5 0.5) t) k) k))
      (* (* (fabs l) (fabs l)) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(l) <= 2e+40) {
		tmp = ((fma(-0.16666666666666666, (fabs(l) / t), (fabs(l) / (pow(k, 2.0) * t))) / pow(k, 2.0)) * fabs(l)) * 2.0;
	} else if (fabs(l) <= 1.65e+186) {
		tmp = (fabs(l) + fabs(l)) * (fabs(l) / (fma((t * fma(0.08611111111111111, (k * k), 0.16666666666666666)), (k * k), t) * pow(k, 4.0)));
	} else {
		tmp = (cos(k) / ((((0.5 - 0.5) * t) * k) * k)) * ((fabs(l) * fabs(l)) * 2.0);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(l) <= 2e+40)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(abs(l) / t), Float64(abs(l) / Float64((k ^ 2.0) * t))) / (k ^ 2.0)) * abs(l)) * 2.0);
	elseif (abs(l) <= 1.65e+186)
		tmp = Float64(Float64(abs(l) + abs(l)) * Float64(abs(l) / Float64(fma(Float64(t * fma(0.08611111111111111, Float64(k * k), 0.16666666666666666)), Float64(k * k), t) * (k ^ 4.0))));
	else
		tmp = Float64(Float64(cos(k) / Float64(Float64(Float64(Float64(0.5 - 0.5) * t) * k) * k)) * Float64(Float64(abs(l) * abs(l)) * 2.0));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[l], $MachinePrecision], 2e+40], N[(N[(N[(N[(-0.16666666666666666 * N[(N[Abs[l], $MachinePrecision] / t), $MachinePrecision] + N[(N[Abs[l], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[N[Abs[l], $MachinePrecision], 1.65e+186], N[(N[(N[Abs[l], $MachinePrecision] + N[Abs[l], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[l], $MachinePrecision] / N[(N[(N[(t * N[(0.08611111111111111 * N[(k * k), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision] + t), $MachinePrecision] * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(N[(0.5 - 0.5), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.00000000000000006e40

   1. Initial program 37.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 31.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 38.5%

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

   6. Taylor expanded in k around 0

      \[\leadsto \left(\color{blue}{\frac{\frac{-1}{6} \cdot \frac{{k}^{2} \cdot \ell}{t} + \frac{\ell}{t}}{{k}^{4}}} \cdot \ell\right) \cdot 2 \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{\frac{-1}{6} \cdot \frac{{k}^{2} \cdot \ell}{t} + \frac{\ell}{t}}{\color{blue}{{k}^{4}}} \cdot \ell\right) \cdot 2 \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{\color{blue}{k}}^{4}} \cdot \ell\right) \cdot 2 \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      7. lower-pow.f64 51.1%

         \[\leadsto \left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{\color{blue}{4}}} \cdot \ell\right) \cdot 2 \]

   8. Applied rewrites 51.1%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}}} \cdot \ell\right) \cdot 2 \]

   9. Taylor expanded in k around inf

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

       1. lower-/.f64 N/A

          \[\leadsto \left(\frac{\frac{-1}{6} \cdot \frac{\ell}{t} + \frac{\ell}{{k}^{2} \cdot t}}{{k}^{\color{blue}{2}}} \cdot \ell\right) \cdot 2 \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       3. lower-/.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       4. lower-/.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       6. lower-pow.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       7. lower-pow.f64 67.4%

          \[\leadsto \left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

   11. Applied rewrites 67.4%

       \[\leadsto \left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{\color{blue}{{k}^{2}}} \cdot \ell\right) \cdot 2 \]

   if 2.00000000000000006e40 < l < 1.65000000000000012e186

   1. Initial program 37.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 31.3%

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

   4. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{4} \cdot \left(t + {k}^{2} \cdot \left(\frac{31}{360} \cdot \left({k}^{2} \cdot t\right) + \frac{1}{6} \cdot t\right)\right)}} \cdot 2 \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot \color{blue}{\left(t + {k}^{2} \cdot \left(\frac{31}{360} \cdot \left({k}^{2} \cdot t\right) + \frac{1}{6} \cdot t\right)\right)}} \cdot 2 \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot \left(\color{blue}{t} + {k}^{2} \cdot \left(\frac{31}{360} \cdot \left({k}^{2} \cdot t\right) + \frac{1}{6} \cdot t\right)\right)} \cdot 2 \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot \left(t + \color{blue}{{k}^{2} \cdot \left(\frac{31}{360} \cdot \left({k}^{2} \cdot t\right) + \frac{1}{6} \cdot t\right)}\right)} \cdot 2 \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot \left(t + {k}^{2} \cdot \color{blue}{\left(\frac{31}{360} \cdot \left({k}^{2} \cdot t\right) + \frac{1}{6} \cdot t\right)}\right)} \cdot 2 \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot \left(t + {k}^{2} \cdot \left(\color{blue}{\frac{31}{360} \cdot \left({k}^{2} \cdot t\right)} + \frac{1}{6} \cdot t\right)\right)} \cdot 2 \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot \left(t + {k}^{2} \cdot \mathsf{fma}\left(\frac{31}{360}, \color{blue}{{k}^{2} \cdot t}, \frac{1}{6} \cdot t\right)\right)} \cdot 2 \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot \left(t + {k}^{2} \cdot \mathsf{fma}\left(\frac{31}{360}, {k}^{2} \cdot \color{blue}{t}, \frac{1}{6} \cdot t\right)\right)} \cdot 2 \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot \left(t + {k}^{2} \cdot \mathsf{fma}\left(\frac{31}{360}, {k}^{2} \cdot t, \frac{1}{6} \cdot t\right)\right)} \cdot 2 \]

      9. lower-*.f64 62.9%

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot \left(t + {k}^{2} \cdot \mathsf{fma}\left(0.08611111111111111, {k}^{2} \cdot t, 0.16666666666666666 \cdot t\right)\right)} \cdot 2 \]

   6. Applied rewrites 62.9%

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{4} \cdot \left(t + {k}^{2} \cdot \mathsf{fma}\left(0.08611111111111111, {k}^{2} \cdot t, 0.16666666666666666 \cdot t\right)\right)}} \cdot 2 \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{{k}^{4} \cdot \left(t + {k}^{2} \cdot \mathsf{fma}\left(\frac{31}{360}, {k}^{2} \cdot t, \frac{1}{6} \cdot t\right)\right)} \cdot 2} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \frac{\ell \cdot \ell}{{k}^{4} \cdot \left(t + {k}^{2} \cdot \mathsf{fma}\left(\frac{31}{360}, {k}^{2} \cdot t, \frac{1}{6} \cdot t\right)\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{{k}^{4} \cdot \left(t + {k}^{2} \cdot \mathsf{fma}\left(\frac{31}{360}, {k}^{2} \cdot t, \frac{1}{6} \cdot t\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot \left(t + {k}^{2} \cdot \mathsf{fma}\left(\frac{31}{360}, {k}^{2} \cdot t, \frac{1}{6} \cdot t\right)\right)} \]

      5. associate-/l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{k}^{4} \cdot \left(t + {k}^{2} \cdot \mathsf{fma}\left(\frac{31}{360}, {k}^{2} \cdot t, \frac{1}{6} \cdot t\right)\right)}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{{k}^{4} \cdot \left(t + {k}^{2} \cdot \mathsf{fma}\left(\frac{31}{360}, {k}^{2} \cdot t, \frac{1}{6} \cdot t\right)\right)}} \]

   8. Applied rewrites 68.9%

      \[\leadsto \color{blue}{\left(\ell + \ell\right) \cdot \frac{\ell}{\mathsf{fma}\left(t \cdot \mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, t\right) \cdot {k}^{4}}} \]

   if 1.65000000000000012e186 < l

   1. Initial program 37.2%

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

   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 74.5%

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

   4. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

   6. Applied rewrites 70.9%

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

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\cos k}{\left(\left(\left(0.5 - \frac{1}{2}\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 37.1%

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

Alternative 8: 74.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-239}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2}}{{k}^{2}}\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 5e-239)
   (*
    (*
     (/ (fma -0.16666666666666666 (/ l t) (/ l (* (pow k 2.0) t))) (pow k 2.0))
     l)
    2.0)
   (* 2.0 (* (/ (cos k) (* (* k k) t)) (/ (pow l 2.0) (pow k 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e-239) {
		tmp = ((fma(-0.16666666666666666, (l / t), (l / (pow(k, 2.0) * t))) / pow(k, 2.0)) * l) * 2.0;
	} else {
		tmp = 2.0 * ((cos(k) / ((k * k) * t)) * (pow(l, 2.0) / pow(k, 2.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 5e-239)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(l / t), Float64(l / Float64((k ^ 2.0) * t))) / (k ^ 2.0)) * l) * 2.0);
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(Float64(k * k) * t)) * Float64((l ^ 2.0) / (k ^ 2.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 5e-239], N[(N[(N[(N[(-0.16666666666666666 * N[(l / t), $MachinePrecision] + N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5e-239

   1. Initial program 37.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 31.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 38.5%

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

   6. Taylor expanded in k around 0

      \[\leadsto \left(\color{blue}{\frac{\frac{-1}{6} \cdot \frac{{k}^{2} \cdot \ell}{t} + \frac{\ell}{t}}{{k}^{4}}} \cdot \ell\right) \cdot 2 \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{\frac{-1}{6} \cdot \frac{{k}^{2} \cdot \ell}{t} + \frac{\ell}{t}}{\color{blue}{{k}^{4}}} \cdot \ell\right) \cdot 2 \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{\color{blue}{k}}^{4}} \cdot \ell\right) \cdot 2 \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      7. lower-pow.f64 51.1%

         \[\leadsto \left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{\color{blue}{4}}} \cdot \ell\right) \cdot 2 \]

   8. Applied rewrites 51.1%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}}} \cdot \ell\right) \cdot 2 \]

   9. Taylor expanded in k around inf

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

       1. lower-/.f64 N/A

          \[\leadsto \left(\frac{\frac{-1}{6} \cdot \frac{\ell}{t} + \frac{\ell}{{k}^{2} \cdot t}}{{k}^{\color{blue}{2}}} \cdot \ell\right) \cdot 2 \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       3. lower-/.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       4. lower-/.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       6. lower-pow.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       7. lower-pow.f64 67.4%

          \[\leadsto \left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

   11. Applied rewrites 67.4%

       \[\leadsto \left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{\color{blue}{{k}^{2}}} \cdot \ell\right) \cdot 2 \]

   if 5e-239 < (*.f64 l l)

   1. Initial program 37.2%

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

   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 74.5%

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

   4. Applied rewrites 74.5%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 75.1%

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lift-sin.f64 N/A

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

      21. lift-sin.f64 N/A

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

      22. sqr-sin-a N/A

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

   6. Applied rewrites 68.2%

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

   7. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 67.6%

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

   9. Applied rewrites 67.6%

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

Alternative 9: 73.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_1 := {k}^{2} \cdot t\\ \mathbf{if}\;\left|\ell\right| \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\left|\ell\right|}{t}, \frac{\left|\ell\right|}{t\_1}\right)}{{k}^{2}} \cdot \left|\ell\right|\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{\left(t\_1 \cdot k\right) \cdot k} \cdot \left(\left(\left|\ell\right| \cdot \left|\ell\right|\right) \cdot 2\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (pow k 2.0) t)))
   (if (<= (fabs l) 2e+16)
     (*
      (*
       (/
        (fma -0.16666666666666666 (/ (fabs l) t) (/ (fabs l) t_1))
        (pow k 2.0))
       (fabs l))
      2.0)
     (* (/ (cos k) (* (* t_1 k) k)) (* (* (fabs l) (fabs l)) 2.0)))))

C

double code(double t, double l, double k) {
	double t_1 = pow(k, 2.0) * t;
	double tmp;
	if (fabs(l) <= 2e+16) {
		tmp = ((fma(-0.16666666666666666, (fabs(l) / t), (fabs(l) / t_1)) / pow(k, 2.0)) * fabs(l)) * 2.0;
	} else {
		tmp = (cos(k) / ((t_1 * k) * k)) * ((fabs(l) * fabs(l)) * 2.0);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64((k ^ 2.0) * t)
	tmp = 0.0
	if (abs(l) <= 2e+16)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(abs(l) / t), Float64(abs(l) / t_1)) / (k ^ 2.0)) * abs(l)) * 2.0);
	else
		tmp = Float64(Float64(cos(k) / Float64(Float64(t_1 * k) * k)) * Float64(Float64(abs(l) * abs(l)) * 2.0));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Power[k, 2.0], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[l], $MachinePrecision], 2e+16], N[(N[(N[(N[(-0.16666666666666666 * N[(N[Abs[l], $MachinePrecision] / t), $MachinePrecision] + N[(N[Abs[l], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[Cos[k], $MachinePrecision] / N[(N[(t$95$1 * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 2e16

   1. Initial program 37.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 31.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 38.5%

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

   6. Taylor expanded in k around 0

      \[\leadsto \left(\color{blue}{\frac{\frac{-1}{6} \cdot \frac{{k}^{2} \cdot \ell}{t} + \frac{\ell}{t}}{{k}^{4}}} \cdot \ell\right) \cdot 2 \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{\frac{-1}{6} \cdot \frac{{k}^{2} \cdot \ell}{t} + \frac{\ell}{t}}{\color{blue}{{k}^{4}}} \cdot \ell\right) \cdot 2 \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{\color{blue}{k}}^{4}} \cdot \ell\right) \cdot 2 \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      7. lower-pow.f64 51.1%

         \[\leadsto \left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{\color{blue}{4}}} \cdot \ell\right) \cdot 2 \]

   8. Applied rewrites 51.1%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}}} \cdot \ell\right) \cdot 2 \]

   9. Taylor expanded in k around inf

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

       1. lower-/.f64 N/A

          \[\leadsto \left(\frac{\frac{-1}{6} \cdot \frac{\ell}{t} + \frac{\ell}{{k}^{2} \cdot t}}{{k}^{\color{blue}{2}}} \cdot \ell\right) \cdot 2 \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       3. lower-/.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       4. lower-/.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       6. lower-pow.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       7. lower-pow.f64 67.4%

          \[\leadsto \left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

   11. Applied rewrites 67.4%

       \[\leadsto \left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{\color{blue}{{k}^{2}}} \cdot \ell\right) \cdot 2 \]

   if 2e16 < l

   1. Initial program 37.2%

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

   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 74.5%

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

   4. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

   6. Applied rewrites 70.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-pow.f64 66.2%

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

   9. Applied rewrites 66.2%

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

Alternative 10: 73.1% accurate, 1.8× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.6 \cdot 10^{+37}:\\ \;\;\;\;\frac{\ell + \ell}{\left|t\right|} \cdot \frac{\ell}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\ell}{\left|t\right|}, \frac{\ell}{{k}^{2} \cdot \left|t\right|}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 1.6e+37)
    (* (/ (+ l l) (fabs t)) (/ l (pow k 4.0)))
    (*
     (*
      (/
       (fma -0.16666666666666666 (/ l (fabs t)) (/ l (* (pow k 2.0) (fabs t))))
       (pow k 2.0))
      l)
     2.0))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 1.6e+37) {
		tmp = ((l + l) / fabs(t)) * (l / pow(k, 4.0));
	} else {
		tmp = ((fma(-0.16666666666666666, (l / fabs(t)), (l / (pow(k, 2.0) * fabs(t)))) / pow(k, 2.0)) * l) * 2.0;
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 1.6e+37)
		tmp = Float64(Float64(Float64(l + l) / abs(t)) * Float64(l / (k ^ 4.0)));
	else
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(l / abs(t)), Float64(l / Float64((k ^ 2.0) * abs(t)))) / (k ^ 2.0)) * l) * 2.0);
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.6e+37], N[(N[(N[(l + l), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.16666666666666666 * N[(l / N[Abs[t], $MachinePrecision]), $MachinePrecision] + N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.60000000000000007e37

   1. Initial program 37.2%

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

   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 63.1%

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

   4. Applied rewrites 63.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 63.1%

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 69.2%

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

   6. Applied rewrites 69.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 69.2%

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

   8. Applied rewrites 69.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 68.7%

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

   10. Applied rewrites 68.7%

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

   if 1.60000000000000007e37 < t

   1. Initial program 37.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 31.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 38.5%

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

   6. Taylor expanded in k around 0

      \[\leadsto \left(\color{blue}{\frac{\frac{-1}{6} \cdot \frac{{k}^{2} \cdot \ell}{t} + \frac{\ell}{t}}{{k}^{4}}} \cdot \ell\right) \cdot 2 \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{\frac{-1}{6} \cdot \frac{{k}^{2} \cdot \ell}{t} + \frac{\ell}{t}}{\color{blue}{{k}^{4}}} \cdot \ell\right) \cdot 2 \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{\color{blue}{k}}^{4}} \cdot \ell\right) \cdot 2 \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      7. lower-pow.f64 51.1%

         \[\leadsto \left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{\color{blue}{4}}} \cdot \ell\right) \cdot 2 \]

   8. Applied rewrites 51.1%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}}} \cdot \ell\right) \cdot 2 \]

   9. Taylor expanded in k around inf

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

       1. lower-/.f64 N/A

          \[\leadsto \left(\frac{\frac{-1}{6} \cdot \frac{\ell}{t} + \frac{\ell}{{k}^{2} \cdot t}}{{k}^{\color{blue}{2}}} \cdot \ell\right) \cdot 2 \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       3. lower-/.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       4. lower-/.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       6. lower-pow.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

       7. lower-pow.f64 67.4%

          \[\leadsto \left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{{k}^{2}} \cdot \ell\right) \cdot 2 \]

   11. Applied rewrites 67.4%

       \[\leadsto \left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\ell}{t}, \frac{\ell}{{k}^{2} \cdot t}\right)}{\color{blue}{{k}^{2}}} \cdot \ell\right) \cdot 2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 70.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 1.05 \cdot 10^{+15}:\\ \;\;\;\;\frac{\ell + \ell}{t} \cdot \frac{\ell}{{\left(\left|k\right|\right)}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot \frac{\ell}{{\left(\left|k\right|\right)}^{2} \cdot t}\right) \cdot \ell\right) \cdot 2\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs k) 1.05e+15)
   (* (/ (+ l l) t) (/ l (pow (fabs k) 4.0)))
   (* (* (* -0.16666666666666666 (/ l (* (pow (fabs k) 2.0) t))) l) 2.0)))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 1.05e+15) {
		tmp = ((l + l) / t) * (l / pow(fabs(k), 4.0));
	} else {
		tmp = ((-0.16666666666666666 * (l / (pow(fabs(k), 2.0) * t))) * l) * 2.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (abs(k) <= 1.05d+15) then
        tmp = ((l + l) / t) * (l / (abs(k) ** 4.0d0))
    else
        tmp = (((-0.16666666666666666d0) * (l / ((abs(k) ** 2.0d0) * t))) * l) * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(k) <= 1.05e+15) {
		tmp = ((l + l) / t) * (l / Math.pow(Math.abs(k), 4.0));
	} else {
		tmp = ((-0.16666666666666666 * (l / (Math.pow(Math.abs(k), 2.0) * t))) * l) * 2.0;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(k) <= 1.05e+15:
		tmp = ((l + l) / t) * (l / math.pow(math.fabs(k), 4.0))
	else:
		tmp = ((-0.16666666666666666 * (l / (math.pow(math.fabs(k), 2.0) * t))) * l) * 2.0
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 1.05e+15)
		tmp = Float64(Float64(Float64(l + l) / t) * Float64(l / (abs(k) ^ 4.0)));
	else
		tmp = Float64(Float64(Float64(-0.16666666666666666 * Float64(l / Float64((abs(k) ^ 2.0) * t))) * l) * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(k) <= 1.05e+15)
		tmp = ((l + l) / t) * (l / (abs(k) ^ 4.0));
	else
		tmp = ((-0.16666666666666666 * (l / ((abs(k) ^ 2.0) * t))) * l) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 1.05e+15], N[(N[(N[(l + l), $MachinePrecision] / t), $MachinePrecision] * N[(l / N[Power[N[Abs[k], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * N[(l / N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.05e15

   1. Initial program 37.2%

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

   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 63.1%

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

   4. Applied rewrites 63.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 63.1%

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 69.2%

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

   6. Applied rewrites 69.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 69.2%

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

   8. Applied rewrites 69.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 68.7%

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

   10. Applied rewrites 68.7%

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

   if 1.05e15 < k

   1. Initial program 37.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 31.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 38.5%

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

   6. Taylor expanded in k around 0

      \[\leadsto \left(\color{blue}{\frac{\frac{-1}{6} \cdot \frac{{k}^{2} \cdot \ell}{t} + \frac{\ell}{t}}{{k}^{4}}} \cdot \ell\right) \cdot 2 \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{\frac{-1}{6} \cdot \frac{{k}^{2} \cdot \ell}{t} + \frac{\ell}{t}}{\color{blue}{{k}^{4}}} \cdot \ell\right) \cdot 2 \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{\color{blue}{k}}^{4}} \cdot \ell\right) \cdot 2 \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}} \cdot \ell\right) \cdot 2 \]

      7. lower-pow.f64 51.1%

         \[\leadsto \left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{\color{blue}{4}}} \cdot \ell\right) \cdot 2 \]

   8. Applied rewrites 51.1%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{{k}^{2} \cdot \ell}{t}, \frac{\ell}{t}\right)}{{k}^{4}}} \cdot \ell\right) \cdot 2 \]

   9. Taylor expanded in k around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{6} \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot t}}\right) \cdot \ell\right) \cdot 2 \]

       2. lower-/.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{6} \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{t}}\right) \cdot \ell\right) \cdot 2 \]

       3. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{6} \cdot \frac{\ell}{{k}^{2} \cdot t}\right) \cdot \ell\right) \cdot 2 \]

       4. lower-pow.f64 30.5%

          \[\leadsto \left(\left(-0.16666666666666666 \cdot \frac{\ell}{{k}^{2} \cdot t}\right) \cdot \ell\right) \cdot 2 \]

   11. Applied rewrites 30.5%

       \[\leadsto \left(\left(-0.16666666666666666 \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}}\right) \cdot \ell\right) \cdot 2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 68.7% accurate, 4.3× speedup

Math

\[\frac{\ell + \ell}{t} \cdot \frac{\ell}{{k}^{4}} \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ (+ l l) t) (/ l (pow k 4.0))))

C

double code(double t, double l, double k) {
	return ((l + l) / t) * (l / pow(k, 4.0));
}

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 = ((l + l) / t) * (l / (k ** 4.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return ((l + l) / t) * (l / Math.pow(k, 4.0));
}

Python

def code(t, l, k):
	return ((l + l) / t) * (l / math.pow(k, 4.0))

Julia

function code(t, l, k)
	return Float64(Float64(Float64(l + l) / t) * Float64(l / (k ^ 4.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((l + l) / t) * (l / (k ^ 4.0));
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(l + l), $MachinePrecision] / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.2%

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

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 63.1%

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

4. Applied rewrites 63.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 63.1%

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

   4. lift-/.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. pow2 N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 69.2%

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

6. Applied rewrites 69.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. count-2-rev N/A

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

   11. lower-+.f64 69.2%

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

8. Applied rewrites 69.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 68.7%

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

10. Applied rewrites 68.7%

    \[\leadsto \frac{\ell + \ell}{t} \cdot \color{blue}{\frac{\ell}{{k}^{4}}} \]
11. Add Preprocessing

Alternative 13: 68.6% accurate, 4.4× speedup

Math

\[\left(\left(\ell + \ell\right) \cdot {k}^{-4}\right) \cdot \frac{\ell}{t} \]

FPCore

(FPCore (t l k) :precision binary64 (* (* (+ l l) (pow k -4.0)) (/ l t)))

C

double code(double t, double l, double k) {
	return ((l + l) * pow(k, -4.0)) * (l / t);
}

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 + l) * (k ** (-4.0d0))) * (l / t)
end function

Java

public static double code(double t, double l, double k) {
	return ((l + l) * Math.pow(k, -4.0)) * (l / t);
}

Python

def code(t, l, k):
	return ((l + l) * math.pow(k, -4.0)) * (l / t)

Julia

function code(t, l, k)
	return Float64(Float64(Float64(l + l) * (k ^ -4.0)) * Float64(l / t))
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((l + l) * (k ^ -4.0)) * (l / t);
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(l + l), $MachinePrecision] * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.2%

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

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 63.1%

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

4. Applied rewrites 63.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 63.1%

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

   4. lift-/.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. pow2 N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 69.2%

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

6. Applied rewrites 69.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. count-2-rev N/A

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

   11. lower-+.f64 69.2%

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

8. Applied rewrites 69.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lift-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. mult-flip N/A

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

   9. lower-*.f64 N/A

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

   10. lift-pow.f64 N/A

       \[\leadsto \left(\left(\ell + \ell\right) \cdot \frac{1}{{k}^{4}}\right) \cdot \frac{\ell}{t} \]

   11. pow-flip N/A

       \[\leadsto \left(\left(\ell + \ell\right) \cdot {k}^{\left(\mathsf{neg}\left(4\right)\right)}\right) \cdot \frac{\ell}{t} \]

   12. lower-pow.f64 N/A

       \[\leadsto \left(\left(\ell + \ell\right) \cdot {k}^{\left(\mathsf{neg}\left(4\right)\right)}\right) \cdot \frac{\ell}{t} \]

   13. metadata-eval 68.6%

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

10. Applied rewrites 68.6%

    \[\leadsto \left(\left(\ell + \ell\right) \cdot {k}^{-4}\right) \cdot \color{blue}{\frac{\ell}{t}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025188 
(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))))