Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.6% → 88.8%

Time: 8.1s

Alternatives: 18

Speedup: 6.6×

• 27.1% 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: 55.6% 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: 88.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \ell \cdot \cos k\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 2 \cdot 10^{-80}:\\ \;\;\;\;\frac{\ell}{\left|t\right| \cdot \mathsf{fma}\left(2, \frac{{\left(\left|t\right|\right)}^{2} \cdot t\_1}{t\_2}, \frac{{k}^{2} \cdot t\_1}{t\_2}\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left|t\right|}{\ell} \cdot \left(\left(\frac{\sin k \cdot \left|t\right|}{\ell} \cdot \left|t\right|\right) \cdot \left(\mathsf{fma}\left(k, \frac{k}{\left|t\right| \cdot \left|t\right|}, 2\right) \cdot \tan k\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)) (t_2 (* l (cos k))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 2e-80)
      (*
       (/
        l
        (*
         (fabs t)
         (fma
          2.0
          (/ (* (pow (fabs t) 2.0) t_1) t_2)
          (/ (* (pow k 2.0) t_1) t_2))))
       2.0)
      (/
       2.0
       (*
        (/ (fabs t) l)
        (*
         (* (/ (* (sin k) (fabs t)) l) (fabs t))
         (* (fma k (/ k (* (fabs t) (fabs t))) 2.0) (tan k)))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double t_2 = l * cos(k);
	double tmp;
	if (fabs(t) <= 2e-80) {
		tmp = (l / (fabs(t) * fma(2.0, ((pow(fabs(t), 2.0) * t_1) / t_2), ((pow(k, 2.0) * t_1) / t_2)))) * 2.0;
	} else {
		tmp = 2.0 / ((fabs(t) / l) * ((((sin(k) * fabs(t)) / l) * fabs(t)) * (fma(k, (k / (fabs(t) * fabs(t))), 2.0) * tan(k))));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	t_2 = Float64(l * cos(k))
	tmp = 0.0
	if (abs(t) <= 2e-80)
		tmp = Float64(Float64(l / Float64(abs(t) * fma(2.0, Float64(Float64((abs(t) ^ 2.0) * t_1) / t_2), Float64(Float64((k ^ 2.0) * t_1) / t_2)))) * 2.0);
	else
		tmp = Float64(2.0 / Float64(Float64(abs(t) / l) * Float64(Float64(Float64(Float64(sin(k) * abs(t)) / l) * abs(t)) * Float64(fma(k, Float64(k / Float64(abs(t) * abs(t))), 2.0) * tan(k)))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 2e-80], N[(N[(l / N[(N[Abs[t], $MachinePrecision] * N[(2.0 * N[(N[(N[Power[N[Abs[t], $MachinePrecision], 2.0], $MachinePrecision] * t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(N[(N[Power[k, 2.0], $MachinePrecision] * t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 / N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[(k * N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.99999999999999992e-80

   1. Initial program 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 68.2

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

   3. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

   7. Applied rewrites 63.5%

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

   8. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

   10. Applied rewrites 76.3%

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

   if 1.99999999999999992e-80 < t

   1. Initial program 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 68.2

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

   3. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

   7. Applied rewrites 71.0%

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

Alternative 2: 88.4% accurate, 1.0× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 2 \cdot 10^{-80}:\\ \;\;\;\;\frac{\ell}{\frac{{k}^{2} \cdot \left(\left|t\right| \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left|t\right|}{\ell} \cdot \left(\left(\frac{\sin k \cdot \left|t\right|}{\ell} \cdot \left|t\right|\right) \cdot \left(\mathsf{fma}\left(k, \frac{k}{\left|t\right| \cdot \left|t\right|}, 2\right) \cdot \tan k\right)\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 2e-80)
    (*
     (/ l (/ (* (pow k 2.0) (* (fabs t) (pow (sin k) 2.0))) (* l (cos k))))
     2.0)
    (/
     2.0
     (*
      (/ (fabs t) l)
      (*
       (* (/ (* (sin k) (fabs t)) l) (fabs t))
       (* (fma k (/ k (* (fabs t) (fabs t))) 2.0) (tan k))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 2e-80) {
		tmp = (l / ((pow(k, 2.0) * (fabs(t) * pow(sin(k), 2.0))) / (l * cos(k)))) * 2.0;
	} else {
		tmp = 2.0 / ((fabs(t) / l) * ((((sin(k) * fabs(t)) / l) * fabs(t)) * (fma(k, (k / (fabs(t) * fabs(t))), 2.0) * tan(k))));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 2e-80)
		tmp = Float64(Float64(l / Float64(Float64((k ^ 2.0) * Float64(abs(t) * (sin(k) ^ 2.0))) / Float64(l * cos(k)))) * 2.0);
	else
		tmp = Float64(2.0 / Float64(Float64(abs(t) / l) * Float64(Float64(Float64(Float64(sin(k) * abs(t)) / l) * abs(t)) * Float64(fma(k, Float64(k / Float64(abs(t) * abs(t))), 2.0) * tan(k)))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 2e-80], N[(N[(l / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 / N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[(k * N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.99999999999999992e-80

   1. Initial program 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 68.2

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

   3. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

   7. Applied rewrites 63.5%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 65.2

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

   10. Applied rewrites 65.2%

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

   if 1.99999999999999992e-80 < t

   1. Initial program 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 68.2

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

   3. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

   7. Applied rewrites 71.0%

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

Alternative 3: 80.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \frac{\sin k \cdot \left|t\right|}{\ell} \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.8 \cdot 10^{-159}:\\ \;\;\;\;\frac{2}{\left(\left|t\right| \cdot \left(\frac{1}{\ell} \cdot \left(t\_1 \cdot \tan k\right)\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left|t\right|}{\ell} \cdot \left(t\_1 \cdot \left(\mathsf{fma}\left(k, \frac{k}{\left|t\right| \cdot \left|t\right|}, 2\right) \cdot \tan k\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ (* (sin k) (fabs t)) l) (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1.8e-159)
      (/ 2.0 (* (* (fabs t) (* (/ 1.0 l) (* t_1 (tan k)))) 2.0))
      (/
       2.0
       (*
        (/ (fabs t) l)
        (* t_1 (* (fma k (/ k (* (fabs t) (fabs t))) 2.0) (tan k)))))))))

C

double code(double t, double l, double k) {
	double t_1 = ((sin(k) * fabs(t)) / l) * fabs(t);
	double tmp;
	if (fabs(t) <= 1.8e-159) {
		tmp = 2.0 / ((fabs(t) * ((1.0 / l) * (t_1 * tan(k)))) * 2.0);
	} else {
		tmp = 2.0 / ((fabs(t) / l) * (t_1 * (fma(k, (k / (fabs(t) * fabs(t))), 2.0) * tan(k))));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(Float64(Float64(sin(k) * abs(t)) / l) * abs(t))
	tmp = 0.0
	if (abs(t) <= 1.8e-159)
		tmp = Float64(2.0 / Float64(Float64(abs(t) * Float64(Float64(1.0 / l) * Float64(t_1 * tan(k)))) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(abs(t) / l) * Float64(t_1 * Float64(fma(k, Float64(k / Float64(abs(t) * abs(t))), 2.0) * tan(k)))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.8e-159], N[(2.0 / N[(N[(N[Abs[t], $MachinePrecision] * N[(N[(1.0 / l), $MachinePrecision] * N[(t$95$1 * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$1 * N[(N[(k * N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.80000000000000011e-159

   1. Initial program 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 68.2

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

   3. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 76.9%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 69.0%

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

   if 1.80000000000000011e-159 < t

   1. Initial program 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 68.2

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

   3. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

   7. Applied rewrites 71.0%

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

Alternative 4: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \frac{\sin k \cdot \left|t\right|}{\ell} \cdot \left|t\right|\\ t_2 := \frac{2}{\left(\left|t\right| \cdot \left(\frac{1}{\ell} \cdot \left(t\_1 \cdot \tan k\right)\right)\right) \cdot 2}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.8 \cdot 10^{-159}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\left|t\right| \leq 10^{+199}:\\ \;\;\;\;\frac{\ell}{\left(\left(\mathsf{fma}\left(\frac{k}{\left|t\right| \cdot \left|t\right|}, k, 2\right) \cdot \tan k\right) \cdot \left|t\right|\right) \cdot t\_1} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ (* (sin k) (fabs t)) l) (fabs t)))
        (t_2 (/ 2.0 (* (* (fabs t) (* (/ 1.0 l) (* t_1 (tan k)))) 2.0))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1.8e-159)
      t_2
      (if (<= (fabs t) 1e+199)
        (*
         (/
          l
          (*
           (* (* (fma (/ k (* (fabs t) (fabs t))) k 2.0) (tan k)) (fabs t))
           t_1))
         2.0)
        t_2)))))

C

double code(double t, double l, double k) {
	double t_1 = ((sin(k) * fabs(t)) / l) * fabs(t);
	double t_2 = 2.0 / ((fabs(t) * ((1.0 / l) * (t_1 * tan(k)))) * 2.0);
	double tmp;
	if (fabs(t) <= 1.8e-159) {
		tmp = t_2;
	} else if (fabs(t) <= 1e+199) {
		tmp = (l / (((fma((k / (fabs(t) * fabs(t))), k, 2.0) * tan(k)) * fabs(t)) * t_1)) * 2.0;
	} else {
		tmp = t_2;
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(Float64(Float64(sin(k) * abs(t)) / l) * abs(t))
	t_2 = Float64(2.0 / Float64(Float64(abs(t) * Float64(Float64(1.0 / l) * Float64(t_1 * tan(k)))) * 2.0))
	tmp = 0.0
	if (abs(t) <= 1.8e-159)
		tmp = t_2;
	elseif (abs(t) <= 1e+199)
		tmp = Float64(Float64(l / Float64(Float64(Float64(fma(Float64(k / Float64(abs(t) * abs(t))), k, 2.0) * tan(k)) * abs(t)) * t_1)) * 2.0);
	else
		tmp = t_2;
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[(N[(N[Abs[t], $MachinePrecision] * N[(N[(1.0 / l), $MachinePrecision] * N[(t$95$1 * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.8e-159], t$95$2, If[LessEqual[N[Abs[t], $MachinePrecision], 1e+199], N[(N[(l / N[(N[(N[(N[(N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], t$95$2]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.80000000000000011e-159 or 1.0000000000000001e199 < t

   1. Initial program 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 68.2

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

   3. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 76.9%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 69.0%

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

   if 1.80000000000000011e-159 < t < 1.0000000000000001e199

   1. Initial program 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 68.2

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

   3. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

   7. Applied rewrites 63.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 68.7

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 68.7

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

   9. Applied rewrites 68.7%

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

Alternative 5: 79.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \frac{\sin k \cdot \left|t\right|}{\ell}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.8 \cdot 10^{-159}:\\ \;\;\;\;\frac{2}{\left(\left|t\right| \cdot \left(\frac{1}{\ell} \cdot \left(\left(t\_1 \cdot \left|t\right|\right) \cdot \tan k\right)\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\mathsf{fma}\left(\frac{k}{\left|t\right| \cdot \left|t\right|}, k, 2\right) \cdot \left(\left(\tan k \cdot \left|t\right|\right) \cdot t\_1\right)} \cdot \frac{2}{\left|t\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (* (sin k) (fabs t)) l)))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1.8e-159)
      (/ 2.0 (* (* (fabs t) (* (/ 1.0 l) (* (* t_1 (fabs t)) (tan k)))) 2.0))
      (*
       (/
        l
        (*
         (fma (/ k (* (fabs t) (fabs t))) k 2.0)
         (* (* (tan k) (fabs t)) t_1)))
       (/ 2.0 (fabs t)))))))

C

double code(double t, double l, double k) {
	double t_1 = (sin(k) * fabs(t)) / l;
	double tmp;
	if (fabs(t) <= 1.8e-159) {
		tmp = 2.0 / ((fabs(t) * ((1.0 / l) * ((t_1 * fabs(t)) * tan(k)))) * 2.0);
	} else {
		tmp = (l / (fma((k / (fabs(t) * fabs(t))), k, 2.0) * ((tan(k) * fabs(t)) * t_1))) * (2.0 / fabs(t));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(Float64(sin(k) * abs(t)) / l)
	tmp = 0.0
	if (abs(t) <= 1.8e-159)
		tmp = Float64(2.0 / Float64(Float64(abs(t) * Float64(Float64(1.0 / l) * Float64(Float64(t_1 * abs(t)) * tan(k)))) * 2.0));
	else
		tmp = Float64(Float64(l / Float64(fma(Float64(k / Float64(abs(t) * abs(t))), k, 2.0) * Float64(Float64(tan(k) * abs(t)) * t_1))) * Float64(2.0 / abs(t)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.8e-159], N[(2.0 / N[(N[(N[Abs[t], $MachinePrecision] * N[(N[(1.0 / l), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.80000000000000011e-159

   1. Initial program 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 68.2

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

   3. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 76.9%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 69.0%

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

   if 1.80000000000000011e-159 < t

   1. Initial program 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 68.2

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

   3. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

   7. Applied rewrites 63.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 69.4%

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

Alternative 6: 78.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{1}{\left|\ell\right|}\\ \mathbf{if}\;\left|\ell\right| \leq 5.6 \cdot 10^{+53}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(t\_1 \cdot \left(\left(\frac{k \cdot t}{\left|\ell\right|} \cdot t\right) \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\left|\ell\right|} \cdot \left(t \cdot \left(\left(\sin k \cdot t\right) \cdot t\_1\right)\right)\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ 1.0 (fabs l))))
   (if (<= (fabs l) 5.6e+53)
     (/
      2.0
      (*
       (* t (* t_1 (* (* (/ (* k t) (fabs l)) t) (tan k))))
       (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
     (/
      2.0
      (* (* (* (/ t (fabs l)) (* t (* (* (sin k) t) t_1))) (tan k)) 2.0)))))

C

double code(double t, double l, double k) {
	double t_1 = 1.0 / fabs(l);
	double tmp;
	if (fabs(l) <= 5.6e+53) {
		tmp = 2.0 / ((t * (t_1 * ((((k * t) / fabs(l)) * t) * tan(k)))) * ((1.0 + pow((k / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / ((((t / fabs(l)) * (t * ((sin(k) * t) * t_1))) * tan(k)) * 2.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 / abs(l)
    if (abs(l) <= 5.6d+53) then
        tmp = 2.0d0 / ((t * (t_1 * ((((k * t) / abs(l)) * t) * tan(k)))) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
    else
        tmp = 2.0d0 / ((((t / abs(l)) * (t * ((sin(k) * t) * t_1))) * tan(k)) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = 1.0 / Math.abs(l);
	double tmp;
	if (Math.abs(l) <= 5.6e+53) {
		tmp = 2.0 / ((t * (t_1 * ((((k * t) / Math.abs(l)) * t) * Math.tan(k)))) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / ((((t / Math.abs(l)) * (t * ((Math.sin(k) * t) * t_1))) * Math.tan(k)) * 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = 1.0 / math.fabs(l)
	tmp = 0
	if math.fabs(l) <= 5.6e+53:
		tmp = 2.0 / ((t * (t_1 * ((((k * t) / math.fabs(l)) * t) * math.tan(k)))) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))
	else:
		tmp = 2.0 / ((((t / math.fabs(l)) * (t * ((math.sin(k) * t) * t_1))) * math.tan(k)) * 2.0)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(1.0 / abs(l))
	tmp = 0.0
	if (abs(l) <= 5.6e+53)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(t_1 * Float64(Float64(Float64(Float64(k * t) / abs(l)) * t) * tan(k)))) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t / abs(l)) * Float64(t * Float64(Float64(sin(k) * t) * t_1))) * tan(k)) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = 1.0 / abs(l);
	tmp = 0.0;
	if (abs(l) <= 5.6e+53)
		tmp = 2.0 / ((t * (t_1 * ((((k * t) / abs(l)) * t) * tan(k)))) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
	else
		tmp = 2.0 / ((((t / abs(l)) * (t * ((sin(k) * t) * t_1))) * tan(k)) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(1.0 / N[Abs[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[l], $MachinePrecision], 5.6e+53], N[(2.0 / N[(N[(t * N[(t$95$1 * N[(N[(N[(N[(k * t), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t / N[Abs[l], $MachinePrecision]), $MachinePrecision] * N[(t * N[(N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 5.6e53

   1. Initial program 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 68.2

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

   3. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 76.9%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 71.6%

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

   if 5.6e53 < l

   1. Initial program 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 68.2

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

   3. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 68.1%

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

Alternative 7: 77.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{1}{\left|\ell\right|}\\ \mathbf{if}\;\left|\ell\right| \leq 260000000:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(t\_1 \cdot \left(\left(\frac{k \cdot t}{\left|\ell\right|} \cdot t\right) \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(t\_1 \cdot \left(\left(\frac{\sin k \cdot t}{\left|\ell\right|} \cdot t\right) \cdot \tan k\right)\right)\right) \cdot 2}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ 1.0 (fabs l))))
   (if (<= (fabs l) 260000000.0)
     (/
      2.0
      (*
       (* t (* t_1 (* (* (/ (* k t) (fabs l)) t) (tan k))))
       (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
     (/
      2.0
      (* (* t (* t_1 (* (* (/ (* (sin k) t) (fabs l)) t) (tan k)))) 2.0)))))

C

double code(double t, double l, double k) {
	double t_1 = 1.0 / fabs(l);
	double tmp;
	if (fabs(l) <= 260000000.0) {
		tmp = 2.0 / ((t * (t_1 * ((((k * t) / fabs(l)) * t) * tan(k)))) * ((1.0 + pow((k / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / ((t * (t_1 * ((((sin(k) * t) / fabs(l)) * t) * tan(k)))) * 2.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 / abs(l)
    if (abs(l) <= 260000000.0d0) then
        tmp = 2.0d0 / ((t * (t_1 * ((((k * t) / abs(l)) * t) * tan(k)))) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
    else
        tmp = 2.0d0 / ((t * (t_1 * ((((sin(k) * t) / abs(l)) * t) * tan(k)))) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = 1.0 / Math.abs(l);
	double tmp;
	if (Math.abs(l) <= 260000000.0) {
		tmp = 2.0 / ((t * (t_1 * ((((k * t) / Math.abs(l)) * t) * Math.tan(k)))) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / ((t * (t_1 * ((((Math.sin(k) * t) / Math.abs(l)) * t) * Math.tan(k)))) * 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = 1.0 / math.fabs(l)
	tmp = 0
	if math.fabs(l) <= 260000000.0:
		tmp = 2.0 / ((t * (t_1 * ((((k * t) / math.fabs(l)) * t) * math.tan(k)))) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))
	else:
		tmp = 2.0 / ((t * (t_1 * ((((math.sin(k) * t) / math.fabs(l)) * t) * math.tan(k)))) * 2.0)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(1.0 / abs(l))
	tmp = 0.0
	if (abs(l) <= 260000000.0)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(t_1 * Float64(Float64(Float64(Float64(k * t) / abs(l)) * t) * tan(k)))) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(t_1 * Float64(Float64(Float64(Float64(sin(k) * t) / abs(l)) * t) * tan(k)))) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = 1.0 / abs(l);
	tmp = 0.0;
	if (abs(l) <= 260000000.0)
		tmp = 2.0 / ((t * (t_1 * ((((k * t) / abs(l)) * t) * tan(k)))) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
	else
		tmp = 2.0 / ((t * (t_1 * ((((sin(k) * t) / abs(l)) * t) * tan(k)))) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(1.0 / N[Abs[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[l], $MachinePrecision], 260000000.0], N[(2.0 / N[(N[(t * N[(t$95$1 * N[(N[(N[(N[(k * t), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(t$95$1 * N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 2.6e8

   1. Initial program 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 68.2

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

   3. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 76.9%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 71.6%

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

   if 2.6e8 < l

   1. Initial program 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 68.2

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

   3. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 76.9%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 69.0%

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

Alternative 8: 72.7% accurate, 1.2× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.45 \cdot 10^{-169}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left|t\right|}{\ell} \cdot \frac{\left(\left|t\right| \cdot \left|t\right|\right) \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left|t\right| \cdot \left(\frac{1}{\ell} \cdot \left(\left(\frac{k \cdot \left|t\right|}{\ell} \cdot \left|t\right|\right) \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{\left|t\right|}\right)}^{2}\right) + 1\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 1.45e-169)
    (/
     2.0
     (*
      (* (* (/ (fabs t) l) (/ (* (* (fabs t) (fabs t)) (sin k)) l)) (tan k))
      2.0))
    (/
     2.0
     (*
      (* (fabs t) (* (/ 1.0 l) (* (* (/ (* k (fabs t)) l) (fabs t)) (tan k))))
      (+ (+ 1.0 (pow (/ k (fabs t)) 2.0)) 1.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 1.45e-169) {
		tmp = 2.0 / ((((fabs(t) / l) * (((fabs(t) * fabs(t)) * sin(k)) / l)) * tan(k)) * 2.0);
	} else {
		tmp = 2.0 / ((fabs(t) * ((1.0 / l) * ((((k * fabs(t)) / l) * fabs(t)) * tan(k)))) * ((1.0 + pow((k / fabs(t)), 2.0)) + 1.0));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(t) <= 1.45e-169) {
		tmp = 2.0 / ((((Math.abs(t) / l) * (((Math.abs(t) * Math.abs(t)) * Math.sin(k)) / l)) * Math.tan(k)) * 2.0);
	} else {
		tmp = 2.0 / ((Math.abs(t) * ((1.0 / l) * ((((k * Math.abs(t)) / l) * Math.abs(t)) * Math.tan(k)))) * ((1.0 + Math.pow((k / Math.abs(t)), 2.0)) + 1.0));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(t) <= 1.45e-169:
		tmp = 2.0 / ((((math.fabs(t) / l) * (((math.fabs(t) * math.fabs(t)) * math.sin(k)) / l)) * math.tan(k)) * 2.0)
	else:
		tmp = 2.0 / ((math.fabs(t) * ((1.0 / l) * ((((k * math.fabs(t)) / l) * math.fabs(t)) * math.tan(k)))) * ((1.0 + math.pow((k / math.fabs(t)), 2.0)) + 1.0))
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 1.45e-169)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(abs(t) / l) * Float64(Float64(Float64(abs(t) * abs(t)) * sin(k)) / l)) * tan(k)) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(abs(t) * Float64(Float64(1.0 / l) * Float64(Float64(Float64(Float64(k * abs(t)) / l) * abs(t)) * tan(k)))) * Float64(Float64(1.0 + (Float64(k / abs(t)) ^ 2.0)) + 1.0)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(t) <= 1.45e-169)
		tmp = 2.0 / ((((abs(t) / l) * (((abs(t) * abs(t)) * sin(k)) / l)) * tan(k)) * 2.0);
	else
		tmp = 2.0 / ((abs(t) * ((1.0 / l) * ((((k * abs(t)) / l) * abs(t)) * tan(k)))) * ((1.0 + ((k / abs(t)) ^ 2.0)) + 1.0));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.45e-169], N[(2.0 / N[(N[(N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Abs[t], $MachinePrecision] * N[(N[(1.0 / l), $MachinePrecision] * N[(N[(N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / N[Abs[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.4500000000000001e-169

   1. Initial program 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 68.2

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

   3. Applied rewrites 68.2%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 62.8%

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

   if 1.4500000000000001e-169 < t

   1. Initial program 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 68.2

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

   3. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 76.9%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 71.6%

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

Alternative 9: 71.6% accurate, 1.2× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.45 \cdot 10^{-169}:\\ \;\;\;\;\frac{\frac{\ell}{\left(\left(\sin k \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left|t\right|} \cdot \ell}{2 \cdot k} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left|t\right| \cdot \left(\frac{1}{\ell} \cdot \left(\left(\frac{k \cdot \left|t\right|}{\ell} \cdot \left|t\right|\right) \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{\left|t\right|}\right)}^{2}\right) + 1\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 1.45e-169)
    (*
     (/ (* (/ l (* (* (* (sin k) (fabs t)) (fabs t)) (fabs t))) l) (* 2.0 k))
     2.0)
    (/
     2.0
     (*
      (* (fabs t) (* (/ 1.0 l) (* (* (/ (* k (fabs t)) l) (fabs t)) (tan k))))
      (+ (+ 1.0 (pow (/ k (fabs t)) 2.0)) 1.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 1.45e-169) {
		tmp = (((l / (((sin(k) * fabs(t)) * fabs(t)) * fabs(t))) * l) / (2.0 * k)) * 2.0;
	} else {
		tmp = 2.0 / ((fabs(t) * ((1.0 / l) * ((((k * fabs(t)) / l) * fabs(t)) * tan(k)))) * ((1.0 + pow((k / fabs(t)), 2.0)) + 1.0));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(t) <= 1.45e-169) {
		tmp = (((l / (((Math.sin(k) * Math.abs(t)) * Math.abs(t)) * Math.abs(t))) * l) / (2.0 * k)) * 2.0;
	} else {
		tmp = 2.0 / ((Math.abs(t) * ((1.0 / l) * ((((k * Math.abs(t)) / l) * Math.abs(t)) * Math.tan(k)))) * ((1.0 + Math.pow((k / Math.abs(t)), 2.0)) + 1.0));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(t) <= 1.45e-169:
		tmp = (((l / (((math.sin(k) * math.fabs(t)) * math.fabs(t)) * math.fabs(t))) * l) / (2.0 * k)) * 2.0
	else:
		tmp = 2.0 / ((math.fabs(t) * ((1.0 / l) * ((((k * math.fabs(t)) / l) * math.fabs(t)) * math.tan(k)))) * ((1.0 + math.pow((k / math.fabs(t)), 2.0)) + 1.0))
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 1.45e-169)
		tmp = Float64(Float64(Float64(Float64(l / Float64(Float64(Float64(sin(k) * abs(t)) * abs(t)) * abs(t))) * l) / Float64(2.0 * k)) * 2.0);
	else
		tmp = Float64(2.0 / Float64(Float64(abs(t) * Float64(Float64(1.0 / l) * Float64(Float64(Float64(Float64(k * abs(t)) / l) * abs(t)) * tan(k)))) * Float64(Float64(1.0 + (Float64(k / abs(t)) ^ 2.0)) + 1.0)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(t) <= 1.45e-169)
		tmp = (((l / (((sin(k) * abs(t)) * abs(t)) * abs(t))) * l) / (2.0 * k)) * 2.0;
	else
		tmp = 2.0 / ((abs(t) * ((1.0 / l) * ((((k * abs(t)) / l) * abs(t)) * tan(k)))) * ((1.0 + ((k / abs(t)) ^ 2.0)) + 1.0));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.45e-169], N[(N[(N[(N[(l / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / N[(2.0 * k), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 / N[(N[(N[Abs[t], $MachinePrecision] * N[(N[(1.0 / l), $MachinePrecision] * N[(N[(N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / N[Abs[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.4500000000000001e-169

   1. Initial program 55.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 55.8%

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

   4. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \color{blue}{\left(k \cdot \left(1 + \frac{1}{3} \cdot {k}^{2}\right)\right)}\right)} \cdot 2 \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 52.6

         \[\leadsto \frac{\ell \cdot \ell}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(k \cdot \left(1 + 0.3333333333333333 \cdot {k}^{\color{blue}{2}}\right)\right)\right)} \cdot 2 \]

   6. Applied rewrites 52.6%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \color{blue}{\left(k \cdot \left(1 + 0.3333333333333333 \cdot {k}^{2}\right)\right)}\right)} \cdot 2 \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

   8. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t} \cdot \ell}{\left(\mathsf{fma}\left(0.3333333333333333, k \cdot k, 1\right) \cdot k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)}} \cdot 2 \]

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 61.5

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

   11. Applied rewrites 61.5%

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

   if 1.4500000000000001e-169 < t

   1. Initial program 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 68.2

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

   3. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 76.9%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 71.6%

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

Alternative 10: 71.1% accurate, 1.3× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.45 \cdot 10^{-169}:\\ \;\;\;\;\frac{\frac{\ell}{\left(\left(\sin k \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left|t\right|} \cdot \ell}{2 \cdot k} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left|t\right|}{\ell} \cdot \left(\left|t\right| \cdot \frac{k \cdot \left|t\right|}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{\left|t\right|}\right)}^{2}\right) + 1\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 1.45e-169)
    (*
     (/ (* (/ l (* (* (* (sin k) (fabs t)) (fabs t)) (fabs t))) l) (* 2.0 k))
     2.0)
    (/
     2.0
     (*
      (* (* (/ (fabs t) l) (* (fabs t) (/ (* k (fabs t)) l))) (tan k))
      (+ (+ 1.0 (pow (/ k (fabs t)) 2.0)) 1.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 1.45e-169) {
		tmp = (((l / (((sin(k) * fabs(t)) * fabs(t)) * fabs(t))) * l) / (2.0 * k)) * 2.0;
	} else {
		tmp = 2.0 / ((((fabs(t) / l) * (fabs(t) * ((k * fabs(t)) / l))) * tan(k)) * ((1.0 + pow((k / fabs(t)), 2.0)) + 1.0));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(t) <= 1.45e-169) {
		tmp = (((l / (((Math.sin(k) * Math.abs(t)) * Math.abs(t)) * Math.abs(t))) * l) / (2.0 * k)) * 2.0;
	} else {
		tmp = 2.0 / ((((Math.abs(t) / l) * (Math.abs(t) * ((k * Math.abs(t)) / l))) * Math.tan(k)) * ((1.0 + Math.pow((k / Math.abs(t)), 2.0)) + 1.0));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(t) <= 1.45e-169:
		tmp = (((l / (((math.sin(k) * math.fabs(t)) * math.fabs(t)) * math.fabs(t))) * l) / (2.0 * k)) * 2.0
	else:
		tmp = 2.0 / ((((math.fabs(t) / l) * (math.fabs(t) * ((k * math.fabs(t)) / l))) * math.tan(k)) * ((1.0 + math.pow((k / math.fabs(t)), 2.0)) + 1.0))
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 1.45e-169)
		tmp = Float64(Float64(Float64(Float64(l / Float64(Float64(Float64(sin(k) * abs(t)) * abs(t)) * abs(t))) * l) / Float64(2.0 * k)) * 2.0);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(abs(t) / l) * Float64(abs(t) * Float64(Float64(k * abs(t)) / l))) * tan(k)) * Float64(Float64(1.0 + (Float64(k / abs(t)) ^ 2.0)) + 1.0)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(t) <= 1.45e-169)
		tmp = (((l / (((sin(k) * abs(t)) * abs(t)) * abs(t))) * l) / (2.0 * k)) * 2.0;
	else
		tmp = 2.0 / ((((abs(t) / l) * (abs(t) * ((k * abs(t)) / l))) * tan(k)) * ((1.0 + ((k / abs(t)) ^ 2.0)) + 1.0));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.45e-169], N[(N[(N[(N[(l / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / N[(2.0 * k), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / N[Abs[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.4500000000000001e-169

   1. Initial program 55.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 55.8%

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

   4. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \color{blue}{\left(k \cdot \left(1 + \frac{1}{3} \cdot {k}^{2}\right)\right)}\right)} \cdot 2 \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 52.6

         \[\leadsto \frac{\ell \cdot \ell}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(k \cdot \left(1 + 0.3333333333333333 \cdot {k}^{\color{blue}{2}}\right)\right)\right)} \cdot 2 \]

   6. Applied rewrites 52.6%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \color{blue}{\left(k \cdot \left(1 + 0.3333333333333333 \cdot {k}^{2}\right)\right)}\right)} \cdot 2 \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

   8. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t} \cdot \ell}{\left(\mathsf{fma}\left(0.3333333333333333, k \cdot k, 1\right) \cdot k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)}} \cdot 2 \]

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 61.5

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

   11. Applied rewrites 61.5%

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

   if 1.4500000000000001e-169 < t

   1. Initial program 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 68.2

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

   3. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 71.2

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

   8. Applied rewrites 71.2%

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

Alternative 11: 69.2% accurate, 1.4× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.8 \cdot 10^{-159}:\\ \;\;\;\;\frac{\frac{\ell}{\left(\left(\sin k \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left|t\right|} \cdot \ell}{2 \cdot k} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\mathsf{fma}\left(k, \frac{k}{\left|t\right| \cdot \left|t\right|}, 2\right) \cdot \tan k\right) \cdot \left(\left(\frac{k \cdot \left|t\right|}{\ell} \cdot \left|t\right|\right) \cdot \left|t\right|\right)} \cdot 2\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 1.8e-159)
    (*
     (/ (* (/ l (* (* (* (sin k) (fabs t)) (fabs t)) (fabs t))) l) (* 2.0 k))
     2.0)
    (*
     (/
      l
      (*
       (* (fma k (/ k (* (fabs t) (fabs t))) 2.0) (tan k))
       (* (* (/ (* k (fabs t)) l) (fabs t)) (fabs t))))
     2.0))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 1.8e-159) {
		tmp = (((l / (((sin(k) * fabs(t)) * fabs(t)) * fabs(t))) * l) / (2.0 * k)) * 2.0;
	} else {
		tmp = (l / ((fma(k, (k / (fabs(t) * fabs(t))), 2.0) * tan(k)) * ((((k * fabs(t)) / l) * fabs(t)) * fabs(t)))) * 2.0;
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 1.8e-159)
		tmp = Float64(Float64(Float64(Float64(l / Float64(Float64(Float64(sin(k) * abs(t)) * abs(t)) * abs(t))) * l) / Float64(2.0 * k)) * 2.0);
	else
		tmp = Float64(Float64(l / Float64(Float64(fma(k, Float64(k / Float64(abs(t) * abs(t))), 2.0) * tan(k)) * Float64(Float64(Float64(Float64(k * abs(t)) / l) * abs(t)) * abs(t)))) * 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], 1.8e-159], N[(N[(N[(N[(l / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / N[(2.0 * k), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(l / N[(N[(N[(k * N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.80000000000000011e-159

   1. Initial program 55.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 55.8%

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

   4. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \color{blue}{\left(k \cdot \left(1 + \frac{1}{3} \cdot {k}^{2}\right)\right)}\right)} \cdot 2 \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 52.6

         \[\leadsto \frac{\ell \cdot \ell}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(k \cdot \left(1 + 0.3333333333333333 \cdot {k}^{\color{blue}{2}}\right)\right)\right)} \cdot 2 \]

   6. Applied rewrites 52.6%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \color{blue}{\left(k \cdot \left(1 + 0.3333333333333333 \cdot {k}^{2}\right)\right)}\right)} \cdot 2 \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

   8. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t} \cdot \ell}{\left(\mathsf{fma}\left(0.3333333333333333, k \cdot k, 1\right) \cdot k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)}} \cdot 2 \]

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 61.5

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

   11. Applied rewrites 61.5%

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

   if 1.80000000000000011e-159 < t

   1. Initial program 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 68.2

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

   3. Applied rewrites 68.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

   7. Applied rewrites 63.5%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 60.1%

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

Alternative 12: 67.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 2 \cdot 10^{+175}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \sin \left(\left|k\right|\right)\right)\right) \cdot \left(2 \cdot \frac{\left|k\right|}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\left(\left(\left|k\right| \cdot t\right) \cdot t\right) \cdot t} \cdot \ell}{\left(\mathsf{fma}\left(0.3333333333333333, \left|k\right| \cdot \left|k\right|, 1\right) \cdot \left|k\right|\right) \cdot \mathsf{fma}\left(\left|k\right|, \frac{\left|k\right|}{t \cdot t}, 2\right)} \cdot 2\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs k) 2e+175)
   (/ 2.0 (* (* t (* (* (/ t l) t) (sin (fabs k)))) (* 2.0 (/ (fabs k) l))))
   (*
    (/
     (* (/ l (* (* (* (fabs k) t) t) t)) l)
     (*
      (* (fma 0.3333333333333333 (* (fabs k) (fabs k)) 1.0) (fabs k))
      (fma (fabs k) (/ (fabs k) (* t t)) 2.0)))
    2.0)))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 2e+175) {
		tmp = 2.0 / ((t * (((t / l) * t) * sin(fabs(k)))) * (2.0 * (fabs(k) / l)));
	} else {
		tmp = (((l / (((fabs(k) * t) * t) * t)) * l) / ((fma(0.3333333333333333, (fabs(k) * fabs(k)), 1.0) * fabs(k)) * fma(fabs(k), (fabs(k) / (t * t)), 2.0))) * 2.0;
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 2e+175)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(Float64(t / l) * t) * sin(abs(k)))) * Float64(2.0 * Float64(abs(k) / l))));
	else
		tmp = Float64(Float64(Float64(Float64(l / Float64(Float64(Float64(abs(k) * t) * t) * t)) * l) / Float64(Float64(fma(0.3333333333333333, Float64(abs(k) * abs(k)), 1.0) * abs(k)) * fma(abs(k), Float64(abs(k) / Float64(t * t)), 2.0))) * 2.0);
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 2e+175], N[(2.0 / N[(N[(t * N[(N[(N[(t / l), $MachinePrecision] * t), $MachinePrecision] * N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[Abs[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / N[(N[(N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(0.3333333333333333 * N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.9999999999999999e175

   1. Initial program 55.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 62.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. lift-pow.f64 N/A

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

      10. pow3 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 70.7

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 65.6

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

   8. Applied rewrites 65.6%

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

   if 1.9999999999999999e175 < k

   1. Initial program 55.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 55.8%

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

   4. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \color{blue}{\left(k \cdot \left(1 + \frac{1}{3} \cdot {k}^{2}\right)\right)}\right)} \cdot 2 \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 52.6

         \[\leadsto \frac{\ell \cdot \ell}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(k \cdot \left(1 + 0.3333333333333333 \cdot {k}^{\color{blue}{2}}\right)\right)\right)} \cdot 2 \]

   6. Applied rewrites 52.6%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \color{blue}{\left(k \cdot \left(1 + 0.3333333333333333 \cdot {k}^{2}\right)\right)}\right)} \cdot 2 \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

   8. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t} \cdot \ell}{\left(\mathsf{fma}\left(0.3333333333333333, k \cdot k, 1\right) \cdot k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)}} \cdot 2 \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{\ell}{\left(\left(\color{blue}{k} \cdot t\right) \cdot t\right) \cdot t} \cdot \ell}{\left(\mathsf{fma}\left(\frac{1}{3}, k \cdot k, 1\right) \cdot k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)} \cdot 2 \]
   10. Step-by-step derivation

   11. Applied rewrites 54.4%

       \[\leadsto \frac{\frac{\ell}{\left(\left(\color{blue}{k} \cdot t\right) \cdot t\right) \cdot t} \cdot \ell}{\left(\mathsf{fma}\left(0.3333333333333333, k \cdot k, 1\right) \cdot k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)} \cdot 2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 65.9% accurate, 1.8× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.8 \cdot 10^{-159}:\\ \;\;\;\;\frac{\frac{\ell}{\left(\left(\sin k \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left|t\right|} \cdot \ell}{2 \cdot k} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\left(\left(k \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left|t\right|} \cdot \ell}{\left(\mathsf{fma}\left(0.3333333333333333, k \cdot k, 1\right) \cdot k\right) \cdot \mathsf{fma}\left(k, \frac{k}{\left|t\right| \cdot \left|t\right|}, 2\right)} \cdot 2\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 1.8e-159)
    (*
     (/ (* (/ l (* (* (* (sin k) (fabs t)) (fabs t)) (fabs t))) l) (* 2.0 k))
     2.0)
    (*
     (/
      (* (/ l (* (* (* k (fabs t)) (fabs t)) (fabs t))) l)
      (*
       (* (fma 0.3333333333333333 (* k k) 1.0) k)
       (fma k (/ k (* (fabs t) (fabs t))) 2.0)))
     2.0))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 1.8e-159) {
		tmp = (((l / (((sin(k) * fabs(t)) * fabs(t)) * fabs(t))) * l) / (2.0 * k)) * 2.0;
	} else {
		tmp = (((l / (((k * fabs(t)) * fabs(t)) * fabs(t))) * l) / ((fma(0.3333333333333333, (k * k), 1.0) * k) * fma(k, (k / (fabs(t) * fabs(t))), 2.0))) * 2.0;
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 1.8e-159)
		tmp = Float64(Float64(Float64(Float64(l / Float64(Float64(Float64(sin(k) * abs(t)) * abs(t)) * abs(t))) * l) / Float64(2.0 * k)) * 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(l / Float64(Float64(Float64(k * abs(t)) * abs(t)) * abs(t))) * l) / Float64(Float64(fma(0.3333333333333333, Float64(k * k), 1.0) * k) * fma(k, Float64(k / Float64(abs(t) * abs(t))), 2.0))) * 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], 1.8e-159], N[(N[(N[(N[(l / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / N[(2.0 * k), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(l / N[(N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(0.3333333333333333 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * N[(k * N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.80000000000000011e-159

   1. Initial program 55.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 55.8%

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

   4. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \color{blue}{\left(k \cdot \left(1 + \frac{1}{3} \cdot {k}^{2}\right)\right)}\right)} \cdot 2 \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 52.6

         \[\leadsto \frac{\ell \cdot \ell}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(k \cdot \left(1 + 0.3333333333333333 \cdot {k}^{\color{blue}{2}}\right)\right)\right)} \cdot 2 \]

   6. Applied rewrites 52.6%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \color{blue}{\left(k \cdot \left(1 + 0.3333333333333333 \cdot {k}^{2}\right)\right)}\right)} \cdot 2 \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

   8. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t} \cdot \ell}{\left(\mathsf{fma}\left(0.3333333333333333, k \cdot k, 1\right) \cdot k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)}} \cdot 2 \]

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 61.5

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

   11. Applied rewrites 61.5%

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

   if 1.80000000000000011e-159 < t

   1. Initial program 55.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 55.8%

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

   4. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \color{blue}{\left(k \cdot \left(1 + \frac{1}{3} \cdot {k}^{2}\right)\right)}\right)} \cdot 2 \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 52.6

         \[\leadsto \frac{\ell \cdot \ell}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(k \cdot \left(1 + 0.3333333333333333 \cdot {k}^{\color{blue}{2}}\right)\right)\right)} \cdot 2 \]

   6. Applied rewrites 52.6%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \color{blue}{\left(k \cdot \left(1 + 0.3333333333333333 \cdot {k}^{2}\right)\right)}\right)} \cdot 2 \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(k \cdot \left(1 + \frac{1}{3} \cdot {k}^{2}\right)\right)\right)} \cdot 2 \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(k \cdot \left(1 + \frac{1}{3} \cdot {k}^{2}\right)\right)\right)}} \cdot 2 \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\left(\frac{\ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \frac{\ell}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(k \cdot \left(1 + \frac{1}{3} \cdot {k}^{2}\right)\right)}\right)} \cdot 2 \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(k \cdot \left(1 + \frac{1}{3} \cdot {k}^{2}\right)\right)}} \cdot 2 \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(k \cdot \left(1 + \frac{1}{3} \cdot {k}^{2}\right)\right)}} \cdot 2 \]

   8. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t} \cdot \ell}{\left(\mathsf{fma}\left(0.3333333333333333, k \cdot k, 1\right) \cdot k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)}} \cdot 2 \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{\ell}{\left(\left(\color{blue}{k} \cdot t\right) \cdot t\right) \cdot t} \cdot \ell}{\left(\mathsf{fma}\left(\frac{1}{3}, k \cdot k, 1\right) \cdot k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)} \cdot 2 \]
   10. Step-by-step derivation

   11. Applied rewrites 54.4%

       \[\leadsto \frac{\frac{\ell}{\left(\left(\color{blue}{k} \cdot t\right) \cdot t\right) \cdot t} \cdot \ell}{\left(\mathsf{fma}\left(0.3333333333333333, k \cdot k, 1\right) \cdot k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)} \cdot 2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 65.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_1 := \left|t\right| \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.05 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{\left|t\right|}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\left(\left(k \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left|t\right|} \cdot \ell}{\left(\mathsf{fma}\left(0.3333333333333333, k \cdot k, 1\right) \cdot k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t\_1}, 2\right)} \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (fabs t) (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1.05e-78)
      (/ (/ (* l (/ l (* k k))) (fabs t)) t_1)
      (*
       (/
        (* (/ l (* (* (* k (fabs t)) (fabs t)) (fabs t))) l)
        (* (* (fma 0.3333333333333333 (* k k) 1.0) k) (fma k (/ k t_1) 2.0)))
       2.0)))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) * fabs(t);
	double tmp;
	if (fabs(t) <= 1.05e-78) {
		tmp = ((l * (l / (k * k))) / fabs(t)) / t_1;
	} else {
		tmp = (((l / (((k * fabs(t)) * fabs(t)) * fabs(t))) * l) / ((fma(0.3333333333333333, (k * k), 1.0) * k) * fma(k, (k / t_1), 2.0))) * 2.0;
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) * abs(t))
	tmp = 0.0
	if (abs(t) <= 1.05e-78)
		tmp = Float64(Float64(Float64(l * Float64(l / Float64(k * k))) / abs(t)) / t_1);
	else
		tmp = Float64(Float64(Float64(Float64(l / Float64(Float64(Float64(k * abs(t)) * abs(t)) * abs(t))) * l) / Float64(Float64(fma(0.3333333333333333, Float64(k * k), 1.0) * k) * fma(k, Float64(k / t_1), 2.0))) * 2.0);
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.05e-78], N[(N[(N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[(N[(l / N[(N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(0.3333333333333333 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * N[(k * N[(k / t$95$1), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.05e-78

   1. Initial program 55.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.7

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{\color{blue}{{t}^{3}}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{{t}^{\color{blue}{3}}} \]

      8. cube-mult N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot \color{blue}{\left(t \cdot t\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot \left(t \cdot \color{blue}{t}\right)} \]

      10. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t}}{\color{blue}{t \cdot t}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t}}{\color{blue}{t \cdot t}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t}}{\color{blue}{t} \cdot t} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t}}{t \cdot t} \]

      14. associate-/l* N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t}}{t \cdot t} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t}}{t \cdot t} \]

      16. lower-/.f64 59.0

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t}}{t \cdot t} \]

      17. lift-pow.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t}}{t \cdot t} \]

      18. unpow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t}}{t \cdot t} \]

      19. lower-*.f64 59.0

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t}}{t \cdot t} \]

   6. Applied rewrites 59.0%

      \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t}}{\color{blue}{t \cdot t}} \]

   if 1.05e-78 < t

   1. Initial program 55.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 55.8%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \tan k\right)} \cdot 2} \]

   4. Taylor expanded in k around 0

      \[\leadsto \frac{\ell \cdot \ell}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \color{blue}{\left(k \cdot \left(1 + \frac{1}{3} \cdot {k}^{2}\right)\right)}\right)} \cdot 2 \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(k \cdot \color{blue}{\left(1 + \frac{1}{3} \cdot {k}^{2}\right)}\right)\right)} \cdot 2 \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(k \cdot \left(1 + \color{blue}{\frac{1}{3} \cdot {k}^{2}}\right)\right)\right)} \cdot 2 \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(k \cdot \left(1 + \frac{1}{3} \cdot \color{blue}{{k}^{2}}\right)\right)\right)} \cdot 2 \]

      4. lower-pow.f64 52.6

         \[\leadsto \frac{\ell \cdot \ell}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(k \cdot \left(1 + 0.3333333333333333 \cdot {k}^{\color{blue}{2}}\right)\right)\right)} \cdot 2 \]

   6. Applied rewrites 52.6%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \color{blue}{\left(k \cdot \left(1 + 0.3333333333333333 \cdot {k}^{2}\right)\right)}\right)} \cdot 2 \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(k \cdot \left(1 + \frac{1}{3} \cdot {k}^{2}\right)\right)\right)}} \cdot 2 \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(k \cdot \left(1 + \frac{1}{3} \cdot {k}^{2}\right)\right)\right)} \cdot 2 \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(k \cdot \left(1 + \frac{1}{3} \cdot {k}^{2}\right)\right)\right)}} \cdot 2 \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\left(\frac{\ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \frac{\ell}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(k \cdot \left(1 + \frac{1}{3} \cdot {k}^{2}\right)\right)}\right)} \cdot 2 \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(k \cdot \left(1 + \frac{1}{3} \cdot {k}^{2}\right)\right)}} \cdot 2 \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(k \cdot \left(1 + \frac{1}{3} \cdot {k}^{2}\right)\right)}} \cdot 2 \]

   8. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot t} \cdot \ell}{\left(\mathsf{fma}\left(0.3333333333333333, k \cdot k, 1\right) \cdot k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)}} \cdot 2 \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{\ell}{\left(\left(\color{blue}{k} \cdot t\right) \cdot t\right) \cdot t} \cdot \ell}{\left(\mathsf{fma}\left(\frac{1}{3}, k \cdot k, 1\right) \cdot k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)} \cdot 2 \]
   10. Step-by-step derivation

   11. Applied rewrites 54.4%

       \[\leadsto \frac{\frac{\ell}{\left(\left(\color{blue}{k} \cdot t\right) \cdot t\right) \cdot t} \cdot \ell}{\left(\mathsf{fma}\left(0.3333333333333333, k \cdot k, 1\right) \cdot k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)} \cdot 2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 65.8% accurate, 3.7× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 2.6 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{\left|t\right| \cdot \left|t\right|}}{\left|t\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left|t\right| \cdot \left(\left|t\right| \cdot \left(k \cdot \left|t\right|\right)\right)\right) \cdot k} \cdot \ell\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 2.6e-36)
    (/ (/ (* l (/ l (* k k))) (* (fabs t) (fabs t))) (fabs t))
    (* (/ l (* (* (fabs t) (* (fabs t) (* k (fabs t)))) k)) l))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 2.6e-36) {
		tmp = ((l * (l / (k * k))) / (fabs(t) * fabs(t))) / fabs(t);
	} else {
		tmp = (l / ((fabs(t) * (fabs(t) * (k * fabs(t)))) * k)) * l;
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(t) <= 2.6e-36) {
		tmp = ((l * (l / (k * k))) / (Math.abs(t) * Math.abs(t))) / Math.abs(t);
	} else {
		tmp = (l / ((Math.abs(t) * (Math.abs(t) * (k * Math.abs(t)))) * k)) * l;
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(t) <= 2.6e-36:
		tmp = ((l * (l / (k * k))) / (math.fabs(t) * math.fabs(t))) / math.fabs(t)
	else:
		tmp = (l / ((math.fabs(t) * (math.fabs(t) * (k * math.fabs(t)))) * k)) * l
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 2.6e-36)
		tmp = Float64(Float64(Float64(l * Float64(l / Float64(k * k))) / Float64(abs(t) * abs(t))) / abs(t));
	else
		tmp = Float64(Float64(l / Float64(Float64(abs(t) * Float64(abs(t) * Float64(k * abs(t)))) * k)) * l);
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(t) <= 2.6e-36)
		tmp = ((l * (l / (k * k))) / (abs(t) * abs(t))) / abs(t);
	else
		tmp = (l / ((abs(t) * (abs(t) * (k * abs(t)))) * k)) * l;
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 2.6e-36], N[(N[(N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(N[Abs[t], $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.6e-36

   1. Initial program 55.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.7

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{\color{blue}{{t}^{3}}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{{t}^{\color{blue}{3}}} \]

      8. pow3 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{\left(t \cdot t\right) \cdot \color{blue}{t}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{\left(t \cdot t\right) \cdot t} \]

      10. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot t}}{\color{blue}{t}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot t}}{\color{blue}{t}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot t}}{t} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot t}}{t} \]

      14. associate-/l* N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t \cdot t}}{t} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t \cdot t}}{t} \]

      16. lower-/.f64 59.0

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t \cdot t}}{t} \]

      17. lift-pow.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t \cdot t}}{t} \]

      18. unpow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t \cdot t}}{t} \]

      19. lower-*.f64 59.0

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t \cdot t}}{t} \]

   6. Applied rewrites 59.0%

      \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t \cdot t}}{\color{blue}{t}} \]

   if 2.6e-36 < t

   1. Initial program 55.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.7

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.9

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 60.1

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lift-*.f64 60.1

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 60.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 60.1

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   8. Applied rewrites 60.1%

      \[\leadsto \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k} \cdot \ell \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(k \cdot t\right)\right)\right) \cdot k} \cdot \ell \]

      9. lower-*.f64 64.4

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(k \cdot t\right)\right)\right) \cdot k} \cdot \ell \]

   10. Applied rewrites 64.4%

       \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(k \cdot t\right)\right)\right) \cdot k} \cdot \ell \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 65.2% accurate, 3.7× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{\left|t\right|}}{\left|t\right| \cdot \left|t\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left|t\right| \cdot \left(\left|t\right| \cdot \left(k \cdot \left|t\right|\right)\right)\right) \cdot k} \cdot \ell\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 4e-7)
    (/ (/ (* l (/ l (* k k))) (fabs t)) (* (fabs t) (fabs t)))
    (* (/ l (* (* (fabs t) (* (fabs t) (* k (fabs t)))) k)) l))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 4e-7) {
		tmp = ((l * (l / (k * k))) / fabs(t)) / (fabs(t) * fabs(t));
	} else {
		tmp = (l / ((fabs(t) * (fabs(t) * (k * fabs(t)))) * k)) * l;
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(t) <= 4e-7) {
		tmp = ((l * (l / (k * k))) / Math.abs(t)) / (Math.abs(t) * Math.abs(t));
	} else {
		tmp = (l / ((Math.abs(t) * (Math.abs(t) * (k * Math.abs(t)))) * k)) * l;
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(t) <= 4e-7:
		tmp = ((l * (l / (k * k))) / math.fabs(t)) / (math.fabs(t) * math.fabs(t))
	else:
		tmp = (l / ((math.fabs(t) * (math.fabs(t) * (k * math.fabs(t)))) * k)) * l
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 4e-7)
		tmp = Float64(Float64(Float64(l * Float64(l / Float64(k * k))) / abs(t)) / Float64(abs(t) * abs(t)));
	else
		tmp = Float64(Float64(l / Float64(Float64(abs(t) * Float64(abs(t) * Float64(k * abs(t)))) * k)) * l);
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(t) <= 4e-7)
		tmp = ((l * (l / (k * k))) / abs(t)) / (abs(t) * abs(t));
	else
		tmp = (l / ((abs(t) * (abs(t) * (k * abs(t)))) * k)) * l;
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 4e-7], N[(N[(N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(N[Abs[t], $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.9999999999999998e-7

   1. Initial program 55.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.7

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{\color{blue}{{t}^{3}}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{{t}^{\color{blue}{3}}} \]

      8. cube-mult N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot \color{blue}{\left(t \cdot t\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot \left(t \cdot \color{blue}{t}\right)} \]

      10. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t}}{\color{blue}{t \cdot t}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t}}{\color{blue}{t \cdot t}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t}}{\color{blue}{t} \cdot t} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t}}{t \cdot t} \]

      14. associate-/l* N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t}}{t \cdot t} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t}}{t \cdot t} \]

      16. lower-/.f64 59.0

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t}}{t \cdot t} \]

      17. lift-pow.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t}}{t \cdot t} \]

      18. unpow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t}}{t \cdot t} \]

      19. lower-*.f64 59.0

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t}}{t \cdot t} \]

   6. Applied rewrites 59.0%

      \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t}}{\color{blue}{t \cdot t}} \]

   if 3.9999999999999998e-7 < t

   1. Initial program 55.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.7

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.9

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 60.1

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lift-*.f64 60.1

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 60.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 60.1

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   8. Applied rewrites 60.1%

      \[\leadsto \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k} \cdot \ell \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(k \cdot t\right)\right)\right) \cdot k} \cdot \ell \]

      9. lower-*.f64 64.4

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(k \cdot t\right)\right)\right) \cdot k} \cdot \ell \]

   10. Applied rewrites 64.4%

       \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(k \cdot t\right)\right)\right) \cdot k} \cdot \ell \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 64.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 2 \cdot 10^{-146}:\\ \;\;\;\;\frac{\ell}{\left(t \cdot \left(t \cdot \left(\left|k\right| \cdot t\right)\right)\right) \cdot \left|k\right|} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(\left|k\right| \cdot \left|k\right|\right)\right)} \cdot \ell\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs k) 2e-146)
   (* (/ l (* (* t (* t (* (fabs k) t))) (fabs k))) l)
   (* (/ l (* (* t t) (* t (* (fabs k) (fabs k))))) l)))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 2e-146) {
		tmp = (l / ((t * (t * (fabs(k) * t))) * fabs(k))) * l;
	} else {
		tmp = (l / ((t * t) * (t * (fabs(k) * fabs(k))))) * l;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (abs(k) <= 2d-146) then
        tmp = (l / ((t * (t * (abs(k) * t))) * abs(k))) * l
    else
        tmp = (l / ((t * t) * (t * (abs(k) * abs(k))))) * l
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(k) <= 2e-146) {
		tmp = (l / ((t * (t * (Math.abs(k) * t))) * Math.abs(k))) * l;
	} else {
		tmp = (l / ((t * t) * (t * (Math.abs(k) * Math.abs(k))))) * l;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(k) <= 2e-146:
		tmp = (l / ((t * (t * (math.fabs(k) * t))) * math.fabs(k))) * l
	else:
		tmp = (l / ((t * t) * (t * (math.fabs(k) * math.fabs(k))))) * l
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 2e-146)
		tmp = Float64(Float64(l / Float64(Float64(t * Float64(t * Float64(abs(k) * t))) * abs(k))) * l);
	else
		tmp = Float64(Float64(l / Float64(Float64(t * t) * Float64(t * Float64(abs(k) * abs(k))))) * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(k) <= 2e-146)
		tmp = (l / ((t * (t * (abs(k) * t))) * abs(k))) * l;
	else
		tmp = (l / ((t * t) * (t * (abs(k) * abs(k))))) * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 2e-146], N[(N[(l / N[(N[(t * N[(t * N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(l / N[(N[(t * t), $MachinePrecision] * N[(t * N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.00000000000000005e-146

   1. Initial program 55.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.7

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.9

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 60.1

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lift-*.f64 60.1

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 60.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 60.1

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   8. Applied rewrites 60.1%

      \[\leadsto \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k} \cdot \ell \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(k \cdot t\right)\right)\right) \cdot k} \cdot \ell \]

      9. lower-*.f64 64.4

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(k \cdot t\right)\right)\right) \cdot k} \cdot \ell \]

   10. Applied rewrites 64.4%

       \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(k \cdot t\right)\right)\right) \cdot k} \cdot \ell \]

   if 2.00000000000000005e-146 < k

   1. Initial program 55.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.7

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.9

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 60.1

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lift-*.f64 60.1

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 60.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 60.1

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   8. Applied rewrites 60.1%

      \[\leadsto \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      8. lower-*.f64 59.1

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

   10. Applied rewrites 59.1%

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 63.7% accurate, 6.6× speedup

Math

\[\frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ l (* (* k (* t t)) (* k t))) l))

C

double code(double t, double l, double k) {
	return (l / ((k * (t * t)) * (k * t))) * l;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l / ((k * (t * t)) * (k * t))) * l
end function

Java

public static double code(double t, double l, double k) {
	return (l / ((k * (t * t)) * (k * t))) * l;
}

Python

def code(t, l, k):
	return (l / ((k * (t * t)) * (k * t))) * l

Julia

function code(t, l, k)
	return Float64(Float64(l / Float64(Float64(k * Float64(t * t)) * Float64(k * t))) * l)
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / ((k * (t * t)) * (k * t))) * l;
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[(N[(k * N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]

Derivation

1. Initial program 55.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   4. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   5. lower-pow.f64 51.7

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 51.7%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   6. lower-/.f64 55.9

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   8. *-commutative N/A

      \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

   9. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

   10. unpow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

   11. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

   12. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

   13. lower-*.f64 60.1

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   14. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   15. pow3 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   16. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   17. lift-*.f64 60.1

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

6. Applied rewrites 60.1%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   3. lower-*.f64 60.1

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

8. Applied rewrites 60.1%

   \[\leadsto \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   5. associate-*l* N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

   6. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   9. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. lower-*.f64 63.7

       \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

10. Applied rewrites 63.7%

    \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025175 
(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))))