Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.6% → 83.5%

Time: 10.4s

Alternatives: 16

Speedup: 6.6×

• 26.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.6% accurate, 1.0× speedup

Math

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

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: 83.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{-72}:\\ \;\;\;\;\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\_m\right) \cdot k\right) \cdot k}\\ \mathbf{elif}\;t\_m \leq 6.1 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)}{\ell}} \cdot \frac{1}{\frac{\mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right) \cdot \tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(k \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8e-72)
    (/
     (* (* (* (cos k) l) l) 2.0)
     (* (* (* (- 0.5 (* (cos (+ k k)) 0.5)) t_m) k) k))
    (if (<= t_m 6.1e+102)
      (*
       (/ 2.0 (/ (* (sin k) (* (* t_m t_m) t_m)) l))
       (/ 1.0 (/ (* (fma (/ k (* t_m t_m)) k 2.0) (tan k)) l)))
      (* (/ l (* (* k t_m) (* k t_m))) (/ l t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8e-72) {
		tmp = (((cos(k) * l) * l) * 2.0) / ((((0.5 - (cos((k + k)) * 0.5)) * t_m) * k) * k);
	} else if (t_m <= 6.1e+102) {
		tmp = (2.0 / ((sin(k) * ((t_m * t_m) * t_m)) / l)) * (1.0 / ((fma((k / (t_m * t_m)), k, 2.0) * tan(k)) / l));
	} else {
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8e-72)
		tmp = Float64(Float64(Float64(Float64(cos(k) * l) * l) * 2.0) / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t_m) * k) * k));
	elseif (t_m <= 6.1e+102)
		tmp = Float64(Float64(2.0 / Float64(Float64(sin(k) * Float64(Float64(t_m * t_m) * t_m)) / l)) * Float64(1.0 / Float64(Float64(fma(Float64(k / Float64(t_m * t_m)), k, 2.0) * tan(k)) / l)));
	else
		tmp = Float64(Float64(l / Float64(Float64(k * t_m) * Float64(k * t_m))) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8e-72], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.1e+102], N[(N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(k * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 7.9999999999999997e-72

   1. Initial program 34.6%

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

   3. Applied rewrites 16.9%

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

   4. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 69.9%

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

   8. Applied rewrites 76.0%

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

   if 7.9999999999999997e-72 < t < 6.1000000000000002e102

   1. Initial program 74.9%

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

   3. Applied rewrites 75.8%

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

   5. Applied rewrites 89.1%

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

   if 6.1000000000000002e102 < t

   1. Initial program 60.1%

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

   2. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow3 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 55.7

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

   4. Applied rewrites 55.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. pow3 N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. unswap-sqr N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-/.f64 87.3

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

   6. Applied rewrites 87.3%

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

Alternative 2: 83.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{-72}:\\ \;\;\;\;\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\_m\right) \cdot k\right) \cdot k}\\ \mathbf{elif}\;t\_m \leq 6.1 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right) \cdot \tan k}{\ell} \cdot \frac{\sin k \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(k \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4e-72)
    (/
     (* (* (* (cos k) l) l) 2.0)
     (* (* (* (- 0.5 (* (cos (+ k k)) 0.5)) t_m) k) k))
    (if (<= t_m 6.1e+102)
      (/
       2.0
       (*
        (/ (* (fma (/ k (* t_m t_m)) k 2.0) (tan k)) l)
        (/ (* (sin k) (* (* t_m t_m) t_m)) l)))
      (* (/ l (* (* k t_m) (* k t_m))) (/ l t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4e-72) {
		tmp = (((cos(k) * l) * l) * 2.0) / ((((0.5 - (cos((k + k)) * 0.5)) * t_m) * k) * k);
	} else if (t_m <= 6.1e+102) {
		tmp = 2.0 / (((fma((k / (t_m * t_m)), k, 2.0) * tan(k)) / l) * ((sin(k) * ((t_m * t_m) * t_m)) / l));
	} else {
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4e-72)
		tmp = Float64(Float64(Float64(Float64(cos(k) * l) * l) * 2.0) / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t_m) * k) * k));
	elseif (t_m <= 6.1e+102)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(k / Float64(t_m * t_m)), k, 2.0) * tan(k)) / l) * Float64(Float64(sin(k) * Float64(Float64(t_m * t_m) * t_m)) / l)));
	else
		tmp = Float64(Float64(l / Float64(Float64(k * t_m) * Float64(k * t_m))) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4e-72], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.1e+102], N[(2.0 / N[(N[(N[(N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(k * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.9999999999999999e-72

   1. Initial program 34.5%

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

   3. Applied rewrites 16.8%

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

   4. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 69.9%

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

   8. Applied rewrites 76.0%

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

   if 3.9999999999999999e-72 < t < 6.1000000000000002e102

   1. Initial program 74.9%

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

   3. Applied rewrites 75.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

   5. Applied rewrites 88.8%

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

   if 6.1000000000000002e102 < t

   1. Initial program 60.1%

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

   2. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow3 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 55.7

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

   4. Applied rewrites 55.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. pow3 N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. unswap-sqr N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-/.f64 87.3

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

   6. Applied rewrites 87.3%

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

Alternative 3: 83.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{-122}:\\ \;\;\;\;\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\_m\right) \cdot k\right) \cdot k}\\ \mathbf{elif}\;t\_m \leq 1.4 \cdot 10^{+145}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{\sin k}{\ell} \cdot t\_m\right) \cdot \left(\frac{t\_m}{\ell} \cdot t\_m\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(k \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4e-122)
    (/
     (* (* (* (cos k) l) l) 2.0)
     (* (* (* (- 0.5 (* (cos (+ k k)) 0.5)) t_m) k) k))
    (if (<= t_m 1.4e+145)
      (/
       2.0
       (*
        (* (* (* (/ (sin k) l) t_m) (* (/ t_m l) t_m)) (tan k))
        (fma (/ k (* t_m t_m)) k 2.0)))
      (* (/ l (* (* k t_m) (* k t_m))) (/ l t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4e-122) {
		tmp = (((cos(k) * l) * l) * 2.0) / ((((0.5 - (cos((k + k)) * 0.5)) * t_m) * k) * k);
	} else if (t_m <= 1.4e+145) {
		tmp = 2.0 / (((((sin(k) / l) * t_m) * ((t_m / l) * t_m)) * tan(k)) * fma((k / (t_m * t_m)), k, 2.0));
	} else {
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4e-122)
		tmp = Float64(Float64(Float64(Float64(cos(k) * l) * l) * 2.0) / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t_m) * k) * k));
	elseif (t_m <= 1.4e+145)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(sin(k) / l) * t_m) * Float64(Float64(t_m / l) * t_m)) * tan(k)) * fma(Float64(k / Float64(t_m * t_m)), k, 2.0)));
	else
		tmp = Float64(Float64(l / Float64(Float64(k * t_m) * Float64(k * t_m))) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4e-122], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.4e+145], N[(2.0 / N[(N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(k * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.00000000000000024e-122

   1. Initial program 29.3%

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

   3. Applied rewrites 7.0%

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

   4. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 70.8%

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

   8. Applied rewrites 77.8%

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

   if 4.00000000000000024e-122 < t < 1.3999999999999999e145

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. unpow3 N/A

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

      15. unpow2 N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 81.0

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

   3. Applied rewrites 81.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 81.0

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

   5. Applied rewrites 81.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-/.f64 83.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.1

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

   7. Applied rewrites 83.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. frac-times N/A

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

      8. pow2 N/A

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

      9. associate-*r/ N/A

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

      10. pow2 N/A

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

      11. associate-+l+ N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. lower-/.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 83.1

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

   9. Applied rewrites 83.1%

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

   if 1.3999999999999999e145 < t

   1. Initial program 63.5%

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

   2. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow3 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 58.4

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

   4. Applied rewrites 58.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. pow3 N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. unswap-sqr N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-/.f64 89.3

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

   6. Applied rewrites 89.3%

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

Alternative 4: 79.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.6 \cdot 10^{-70}:\\ \;\;\;\;\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\_m\right) \cdot k\right) \cdot k}\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{+123}:\\ \;\;\;\;\frac{\ell}{\left(t\_m \cdot t\_m\right) \cdot \left(t\_m \cdot k\right)} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(k \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.6e-70)
    (/
     (* (* (* (cos k) l) l) 2.0)
     (* (* (* (- 0.5 (* (cos (+ k k)) 0.5)) t_m) k) k))
    (if (<= t_m 2e+123)
      (* (/ l (* (* t_m t_m) (* t_m k))) (/ l k))
      (* (/ l (* (* k t_m) (* k t_m))) (/ l t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.6e-70) {
		tmp = (((cos(k) * l) * l) * 2.0) / ((((0.5 - (cos((k + k)) * 0.5)) * t_m) * k) * k);
	} else if (t_m <= 2e+123) {
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k);
	} else {
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
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_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 5.6d-70) then
        tmp = (((cos(k) * l) * l) * 2.0d0) / ((((0.5d0 - (cos((k + k)) * 0.5d0)) * t_m) * k) * k)
    else if (t_m <= 2d+123) then
        tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k)
    else
        tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.6e-70) {
		tmp = (((Math.cos(k) * l) * l) * 2.0) / ((((0.5 - (Math.cos((k + k)) * 0.5)) * t_m) * k) * k);
	} else if (t_m <= 2e+123) {
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k);
	} else {
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 5.6e-70:
		tmp = (((math.cos(k) * l) * l) * 2.0) / ((((0.5 - (math.cos((k + k)) * 0.5)) * t_m) * k) * k)
	elif t_m <= 2e+123:
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k)
	else:
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.6e-70)
		tmp = Float64(Float64(Float64(Float64(cos(k) * l) * l) * 2.0) / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t_m) * k) * k));
	elseif (t_m <= 2e+123)
		tmp = Float64(Float64(l / Float64(Float64(t_m * t_m) * Float64(t_m * k))) * Float64(l / k));
	else
		tmp = Float64(Float64(l / Float64(Float64(k * t_m) * Float64(k * t_m))) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 5.6e-70)
		tmp = (((cos(k) * l) * l) * 2.0) / ((((0.5 - (cos((k + k)) * 0.5)) * t_m) * k) * k);
	elseif (t_m <= 2e+123)
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k);
	else
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.6e-70], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2e+123], N[(N[(l / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(k * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.5999999999999998e-70

   1. Initial program 34.7%

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

   3. Applied rewrites 17.2%

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

   4. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 69.9%

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

   8. Applied rewrites 75.9%

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

   if 5.5999999999999998e-70 < t < 1.99999999999999996e123

   1. Initial program 71.9%

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

   3. Applied rewrites 72.7%

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

   4. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. pow3 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 63.0

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

   6. Applied rewrites 63.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. pow3 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 66.7

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

   8. Applied rewrites 66.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 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. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   10. Applied rewrites 73.9%

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

   if 1.99999999999999996e123 < t

   1. Initial program 61.6%

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

   2. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow3 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 56.9

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

   4. Applied rewrites 56.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. pow3 N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. unswap-sqr N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-/.f64 88.3

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

   6. Applied rewrites 88.3%

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

Alternative 5: 77.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{+23}:\\ \;\;\;\;\frac{2}{\left(\left(\left(k \cdot \frac{t\_m}{\ell}\right) \cdot \left(\frac{t\_m}{\ell} \cdot t\_m\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\_m\right) \cdot k} \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.4e+23)
    (/
     2.0
     (*
      (* (* (* k (/ t_m l)) (* (/ t_m l) t_m)) (tan k))
      (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
    (*
     (/ 2.0 (* (* (- 0.5 (* (cos (+ k k)) 0.5)) t_m) k))
     (/ (* (* (cos k) l) l) k)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.4e+23) {
		tmp = 2.0 / ((((k * (t_m / l)) * ((t_m / l) * t_m)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else {
		tmp = (2.0 / (((0.5 - (cos((k + k)) * 0.5)) * t_m) * k)) * (((cos(k) * l) * l) / k);
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
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_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.4d+23) then
        tmp = 2.0d0 / ((((k * (t_m / l)) * ((t_m / l) * t_m)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    else
        tmp = (2.0d0 / (((0.5d0 - (cos((k + k)) * 0.5d0)) * t_m) * k)) * (((cos(k) * l) * l) / k)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.4e+23) {
		tmp = 2.0 / ((((k * (t_m / l)) * ((t_m / l) * t_m)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	} else {
		tmp = (2.0 / (((0.5 - (Math.cos((k + k)) * 0.5)) * t_m) * k)) * (((Math.cos(k) * l) * l) / k);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 3.4e+23:
		tmp = 2.0 / ((((k * (t_m / l)) * ((t_m / l) * t_m)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	else:
		tmp = (2.0 / (((0.5 - (math.cos((k + k)) * 0.5)) * t_m) * k)) * (((math.cos(k) * l) * l) / k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3.4e+23)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * Float64(t_m / l)) * Float64(Float64(t_m / l) * t_m)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t_m) * k)) * Float64(Float64(Float64(cos(k) * l) * l) / k));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 3.4e+23)
		tmp = 2.0 / ((((k * (t_m / l)) * ((t_m / l) * t_m)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	else
		tmp = (2.0 / (((0.5 - (cos((k + k)) * 0.5)) * t_m) * k)) * (((cos(k) * l) * l) / k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 3.4e+23], N[(2.0 / N[(N[(N[(N[(k * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.39999999999999992e23

   1. Initial program 56.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. unpow3 N/A

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

      15. unpow2 N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 68.2

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

   3. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 69.2

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

   5. Applied rewrites 69.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-/.f64 75.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 75.0

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

   7. Applied rewrites 75.0%

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

   8. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 73.0

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

   10. Applied rewrites 73.0%

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

   if 3.39999999999999992e23 < k

   1. Initial program 49.1%

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

   3. Applied rewrites 48.8%

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

   4. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 69.3%

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

   8. Applied rewrites 76.4%

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

Alternative 6: 73.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.4 \cdot 10^{-70}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot t\_m}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{+123}:\\ \;\;\;\;\frac{\ell}{\left(t\_m \cdot t\_m\right) \cdot \left(t\_m \cdot k\right)} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(k \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.4e-70)
    (/ 2.0 (* (/ (* (* k k) t_m) (* l l)) (* (sin k) (tan k))))
    (if (<= t_m 2e+123)
      (* (/ l (* (* t_m t_m) (* t_m k))) (/ l k))
      (* (/ l (* (* k t_m) (* k t_m))) (/ l t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.4e-70) {
		tmp = 2.0 / ((((k * k) * t_m) / (l * l)) * (sin(k) * tan(k)));
	} else if (t_m <= 2e+123) {
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k);
	} else {
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
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_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 5.4d-70) then
        tmp = 2.0d0 / ((((k * k) * t_m) / (l * l)) * (sin(k) * tan(k)))
    else if (t_m <= 2d+123) then
        tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k)
    else
        tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.4e-70) {
		tmp = 2.0 / ((((k * k) * t_m) / (l * l)) * (Math.sin(k) * Math.tan(k)));
	} else if (t_m <= 2e+123) {
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k);
	} else {
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 5.4e-70:
		tmp = 2.0 / ((((k * k) * t_m) / (l * l)) * (math.sin(k) * math.tan(k)))
	elif t_m <= 2e+123:
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k)
	else:
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.4e-70)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * t_m) / Float64(l * l)) * Float64(sin(k) * tan(k))));
	elseif (t_m <= 2e+123)
		tmp = Float64(Float64(l / Float64(Float64(t_m * t_m) * Float64(t_m * k))) * Float64(l / k));
	else
		tmp = Float64(Float64(l / Float64(Float64(k * t_m) * Float64(k * t_m))) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 5.4e-70)
		tmp = 2.0 / ((((k * k) * t_m) / (l * l)) * (sin(k) * tan(k)));
	elseif (t_m <= 2e+123)
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k);
	else
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.4e-70], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2e+123], N[(N[(l / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(k * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.4000000000000003e-70

   1. Initial program 34.7%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. quot-tan N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-tan.f64 72.4

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

   4. Applied rewrites 72.4%

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

   if 5.4000000000000003e-70 < t < 1.99999999999999996e123

   1. Initial program 71.9%

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

   3. Applied rewrites 72.7%

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

   4. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. pow3 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 63.0

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

   6. Applied rewrites 63.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. pow3 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 66.7

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

   8. Applied rewrites 66.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 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. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   10. Applied rewrites 73.9%

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

   if 1.99999999999999996e123 < t

   1. Initial program 61.6%

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

   2. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow3 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 56.9

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

   4. Applied rewrites 56.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. pow3 N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. unswap-sqr N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-/.f64 88.3

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

   6. Applied rewrites 88.3%

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

Alternative 7: 73.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.4 \cdot 10^{-209}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot 2}\\ \mathbf{elif}\;t\_m \leq 4.1 \cdot 10^{-57}:\\ \;\;\;\;\frac{2}{\left(\left(\left(k \cdot \frac{t\_m}{\ell}\right) \cdot \left(\frac{t\_m}{\ell} \cdot t\_m\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{+123}:\\ \;\;\;\;\frac{\ell}{\left(t\_m \cdot t\_m\right) \cdot \left(t\_m \cdot k\right)} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(k \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.4e-209)
    (/ 2.0 (* (* (* (/ k l) (* (* t_m t_m) (/ t_m l))) (tan k)) 2.0))
    (if (<= t_m 4.1e-57)
      (/
       2.0
       (*
        (* (* (* k (/ t_m l)) (* (/ t_m l) t_m)) (tan k))
        (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
      (if (<= t_m 2e+123)
        (* (/ l (* (* t_m t_m) (* t_m k))) (/ l k))
        (* (/ l (* (* k t_m) (* k t_m))) (/ l t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.4e-209) {
		tmp = 2.0 / ((((k / l) * ((t_m * t_m) * (t_m / l))) * tan(k)) * 2.0);
	} else if (t_m <= 4.1e-57) {
		tmp = 2.0 / ((((k * (t_m / l)) * ((t_m / l) * t_m)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else if (t_m <= 2e+123) {
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k);
	} else {
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
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_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 7.4d-209) then
        tmp = 2.0d0 / ((((k / l) * ((t_m * t_m) * (t_m / l))) * tan(k)) * 2.0d0)
    else if (t_m <= 4.1d-57) then
        tmp = 2.0d0 / ((((k * (t_m / l)) * ((t_m / l) * t_m)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    else if (t_m <= 2d+123) then
        tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k)
    else
        tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.4e-209) {
		tmp = 2.0 / ((((k / l) * ((t_m * t_m) * (t_m / l))) * Math.tan(k)) * 2.0);
	} else if (t_m <= 4.1e-57) {
		tmp = 2.0 / ((((k * (t_m / l)) * ((t_m / l) * t_m)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	} else if (t_m <= 2e+123) {
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k);
	} else {
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 7.4e-209:
		tmp = 2.0 / ((((k / l) * ((t_m * t_m) * (t_m / l))) * math.tan(k)) * 2.0)
	elif t_m <= 4.1e-57:
		tmp = 2.0 / ((((k * (t_m / l)) * ((t_m / l) * t_m)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	elif t_m <= 2e+123:
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k)
	else:
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.4e-209)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * Float64(Float64(t_m * t_m) * Float64(t_m / l))) * tan(k)) * 2.0));
	elseif (t_m <= 4.1e-57)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * Float64(t_m / l)) * Float64(Float64(t_m / l) * t_m)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	elseif (t_m <= 2e+123)
		tmp = Float64(Float64(l / Float64(Float64(t_m * t_m) * Float64(t_m * k))) * Float64(l / k));
	else
		tmp = Float64(Float64(l / Float64(Float64(k * t_m) * Float64(k * t_m))) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 7.4e-209)
		tmp = 2.0 / ((((k / l) * ((t_m * t_m) * (t_m / l))) * tan(k)) * 2.0);
	elseif (t_m <= 4.1e-57)
		tmp = 2.0 / ((((k * (t_m / l)) * ((t_m / l) * t_m)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	elseif (t_m <= 2e+123)
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k);
	else
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.4e-209], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.1e-57], N[(2.0 / N[(N[(N[(N[(k * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2e+123], N[(N[(l / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(k * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 7.3999999999999995e-209

   1. Initial program 26.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. unpow3 N/A

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

      15. unpow2 N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 33.1

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

   3. Applied rewrites 33.1%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 55.3%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 50.3%

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

   if 7.3999999999999995e-209 < t < 4.1000000000000001e-57

   1. Initial program 43.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. unpow3 N/A

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

      15. unpow2 N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 60.3

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

   3. Applied rewrites 60.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 62.7

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

   5. Applied rewrites 62.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-/.f64 66.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 66.1

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

   7. Applied rewrites 66.1%

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

   8. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 66.4

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

   10. Applied rewrites 66.4%

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

   if 4.1000000000000001e-57 < t < 1.99999999999999996e123

   1. Initial program 72.0%

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

   3. Applied rewrites 72.9%

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

   4. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. pow3 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 63.3

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

   6. Applied rewrites 63.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. pow3 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 67.3

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

   8. Applied rewrites 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 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. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   10. Applied rewrites 75.0%

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

   if 1.99999999999999996e123 < t

   1. Initial program 61.6%

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

   2. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow3 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 56.9

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

   4. Applied rewrites 56.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. pow3 N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. unswap-sqr N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-/.f64 88.3

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

   6. Applied rewrites 88.3%

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

Alternative 8: 72.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{-209}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot 2}\\ \mathbf{elif}\;t\_m \leq 4.1 \cdot 10^{-57}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{\sin k}{\ell} \cdot t\_m\right) \cdot \left(\frac{t\_m}{\ell} \cdot t\_m\right)\right) \cdot k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{+123}:\\ \;\;\;\;\frac{\ell}{\left(t\_m \cdot t\_m\right) \cdot \left(t\_m \cdot k\right)} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(k \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8e-209)
    (/ 2.0 (* (* (* (/ k l) (* (* t_m t_m) (/ t_m l))) (tan k)) 2.0))
    (if (<= t_m 4.1e-57)
      (/
       2.0
       (*
        (* (* (* (/ (sin k) l) t_m) (* (/ t_m l) t_m)) k)
        (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
      (if (<= t_m 2e+123)
        (* (/ l (* (* t_m t_m) (* t_m k))) (/ l k))
        (* (/ l (* (* k t_m) (* k t_m))) (/ l t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8e-209) {
		tmp = 2.0 / ((((k / l) * ((t_m * t_m) * (t_m / l))) * tan(k)) * 2.0);
	} else if (t_m <= 4.1e-57) {
		tmp = 2.0 / (((((sin(k) / l) * t_m) * ((t_m / l) * t_m)) * k) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else if (t_m <= 2e+123) {
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k);
	} else {
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
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_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 8d-209) then
        tmp = 2.0d0 / ((((k / l) * ((t_m * t_m) * (t_m / l))) * tan(k)) * 2.0d0)
    else if (t_m <= 4.1d-57) then
        tmp = 2.0d0 / (((((sin(k) / l) * t_m) * ((t_m / l) * t_m)) * k) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    else if (t_m <= 2d+123) then
        tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k)
    else
        tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8e-209) {
		tmp = 2.0 / ((((k / l) * ((t_m * t_m) * (t_m / l))) * Math.tan(k)) * 2.0);
	} else if (t_m <= 4.1e-57) {
		tmp = 2.0 / (((((Math.sin(k) / l) * t_m) * ((t_m / l) * t_m)) * k) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	} else if (t_m <= 2e+123) {
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k);
	} else {
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 8e-209:
		tmp = 2.0 / ((((k / l) * ((t_m * t_m) * (t_m / l))) * math.tan(k)) * 2.0)
	elif t_m <= 4.1e-57:
		tmp = 2.0 / (((((math.sin(k) / l) * t_m) * ((t_m / l) * t_m)) * k) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	elif t_m <= 2e+123:
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k)
	else:
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8e-209)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * Float64(Float64(t_m * t_m) * Float64(t_m / l))) * tan(k)) * 2.0));
	elseif (t_m <= 4.1e-57)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(sin(k) / l) * t_m) * Float64(Float64(t_m / l) * t_m)) * k) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	elseif (t_m <= 2e+123)
		tmp = Float64(Float64(l / Float64(Float64(t_m * t_m) * Float64(t_m * k))) * Float64(l / k));
	else
		tmp = Float64(Float64(l / Float64(Float64(k * t_m) * Float64(k * t_m))) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 8e-209)
		tmp = 2.0 / ((((k / l) * ((t_m * t_m) * (t_m / l))) * tan(k)) * 2.0);
	elseif (t_m <= 4.1e-57)
		tmp = 2.0 / (((((sin(k) / l) * t_m) * ((t_m / l) * t_m)) * k) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	elseif (t_m <= 2e+123)
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k);
	else
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8e-209], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.1e-57], N[(2.0 / N[(N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2e+123], N[(N[(l / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(k * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 8.0000000000000004e-209

   1. Initial program 26.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. unpow3 N/A

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

      15. unpow2 N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 33.1

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

   3. Applied rewrites 33.1%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 55.3%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 50.3%

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

   if 8.0000000000000004e-209 < t < 4.1000000000000001e-57

   1. Initial program 43.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. unpow3 N/A

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

      15. unpow2 N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 60.3

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

   3. Applied rewrites 60.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 62.7

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

   5. Applied rewrites 62.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-/.f64 66.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 66.1

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

   7. Applied rewrites 66.1%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 64.0%

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

   if 4.1000000000000001e-57 < t < 1.99999999999999996e123

   1. Initial program 72.0%

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

   3. Applied rewrites 72.9%

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

   4. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. pow3 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 63.3

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

   6. Applied rewrites 63.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. pow3 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 67.3

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

   8. Applied rewrites 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 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. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   10. Applied rewrites 75.0%

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

   if 1.99999999999999996e123 < t

   1. Initial program 61.6%

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

   2. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow3 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 56.9

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

   4. Applied rewrites 56.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. pow3 N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. unswap-sqr N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-/.f64 88.3

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

   6. Applied rewrites 88.3%

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

Alternative 9: 72.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-63}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot 2}\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{+123}:\\ \;\;\;\;\frac{\ell}{\left(t\_m \cdot t\_m\right) \cdot \left(t\_m \cdot k\right)} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(k \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9e-63)
    (/ 2.0 (* (* (* (/ k l) (* (* t_m t_m) (/ t_m l))) (tan k)) 2.0))
    (if (<= t_m 2e+123)
      (* (/ l (* (* t_m t_m) (* t_m k))) (/ l k))
      (* (/ l (* (* k t_m) (* k t_m))) (/ l t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9e-63) {
		tmp = 2.0 / ((((k / l) * ((t_m * t_m) * (t_m / l))) * tan(k)) * 2.0);
	} else if (t_m <= 2e+123) {
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k);
	} else {
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
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_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 9d-63) then
        tmp = 2.0d0 / ((((k / l) * ((t_m * t_m) * (t_m / l))) * tan(k)) * 2.0d0)
    else if (t_m <= 2d+123) then
        tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k)
    else
        tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9e-63) {
		tmp = 2.0 / ((((k / l) * ((t_m * t_m) * (t_m / l))) * Math.tan(k)) * 2.0);
	} else if (t_m <= 2e+123) {
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k);
	} else {
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 9e-63:
		tmp = 2.0 / ((((k / l) * ((t_m * t_m) * (t_m / l))) * math.tan(k)) * 2.0)
	elif t_m <= 2e+123:
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k)
	else:
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 9e-63)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * Float64(Float64(t_m * t_m) * Float64(t_m / l))) * tan(k)) * 2.0));
	elseif (t_m <= 2e+123)
		tmp = Float64(Float64(l / Float64(Float64(t_m * t_m) * Float64(t_m * k))) * Float64(l / k));
	else
		tmp = Float64(Float64(l / Float64(Float64(k * t_m) * Float64(k * t_m))) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 9e-63)
		tmp = 2.0 / ((((k / l) * ((t_m * t_m) * (t_m / l))) * tan(k)) * 2.0);
	elseif (t_m <= 2e+123)
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k);
	else
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9e-63], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2e+123], N[(N[(l / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(k * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 8.9999999999999999e-63

   1. Initial program 35.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. unpow3 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\sin k}{\ell} \cdot \frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\sin k}{\ell} \cdot \frac{\color{blue}{{t}^{2}} \cdot t}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. associate-/l* N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\sin k}{\ell} \cdot \color{blue}{\left({t}^{2} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\sin k}{\ell} \cdot \color{blue}{\left({t}^{2} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      18. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\sin k}{\ell} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\sin k}{\ell} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      20. lower-/.f64 48.5

          \[\leadsto \frac{2}{\left(\left(\frac{\sin k}{\ell} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 48.5%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{\sin k}{\ell} \cdot \left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\left(\frac{\sin k}{\ell} \cdot \left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \color{blue}{2}} \]
   5. Step-by-step derivation

   6. Applied rewrites 51.7%

      \[\leadsto \frac{2}{\left(\left(\frac{\sin k}{\ell} \cdot \left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \color{blue}{2}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{k}}{\ell} \cdot \left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot 2} \]
   8. Step-by-step derivation

   9. Applied rewrites 51.9%

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{k}}{\ell} \cdot \left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot 2} \]

   if 8.9999999999999999e-63 < t < 1.99999999999999996e123

   1. Initial program 72.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 73.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\tan k \cdot \left(\mathsf{fma}\left(k, \frac{k}{t \cdot t}, 1\right) + 1\right)\right)}{\ell \cdot \ell}}} \]

   4. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      3. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      8. associate-*r/ N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      9. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \color{blue}{\ell} \]

      11. lower-*.f64 63.2

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \color{blue}{\ell} \]

   6. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      5. pow3 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)} \cdot \ell \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)} \cdot \ell \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)} \cdot \ell \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)} \cdot \ell \]

      10. pow3 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      12. lift-*.f64 67.1

          \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 67.1%

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \color{blue}{\ell} \]

      2. lift-/.f64 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. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. associate-*l/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      9. times-frac N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]

   10. Applied rewrites 74.5%

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\frac{\ell}{k}} \]

   if 1.99999999999999996e123 < t

   1. Initial program 61.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      2. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. unpow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      8. unpow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      9. unpow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

      11. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      12. lower-*.f64 56.9

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   4. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      9. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      10. unpow3 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      13. times-frac N/A

          \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{2}} \cdot \color{blue}{\frac{\ell}{t}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{2}} \cdot \color{blue}{\frac{\ell}{t}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{2}} \cdot \frac{\color{blue}{\ell}}{t} \]

      16. pow2 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{2}} \cdot \frac{\ell}{t} \]

      17. pow2 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{t} \]

      18. unswap-sqr N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{t} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{t} \]

      20. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{t} \]

      21. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{t} \]

      22. lower-/.f64 88.3

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\color{blue}{t}} \]

   6. Applied rewrites 88.3%

      \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\frac{\ell}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 71.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(t\_m \cdot t\_m\right) \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_2, \left|t\_m\right|\right), k \cdot k, t\_2 \cdot 2\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{+123}:\\ \;\;\;\;\frac{\ell}{\left(t\_m \cdot t\_m\right) \cdot \left(t\_m \cdot k\right)} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(k \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (* t_m t_m) t_m)))
   (*
    t_s
    (if (<= t_m 2.6e-69)
      (/
       2.0
       (*
        (/
         (fma (fma 0.3333333333333333 t_2 (fabs t_m)) (* k k) (* t_2 2.0))
         (* l l))
        (* k k)))
      (if (<= t_m 2e+123)
        (* (/ l (* (* t_m t_m) (* t_m k))) (/ l k))
        (* (/ l (* (* k t_m) (* k t_m))) (/ l t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m * t_m) * t_m;
	double tmp;
	if (t_m <= 2.6e-69) {
		tmp = 2.0 / ((fma(fma(0.3333333333333333, t_2, fabs(t_m)), (k * k), (t_2 * 2.0)) / (l * l)) * (k * k));
	} else if (t_m <= 2e+123) {
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k);
	} else {
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(t_m * t_m) * t_m)
	tmp = 0.0
	if (t_m <= 2.6e-69)
		tmp = Float64(2.0 / Float64(Float64(fma(fma(0.3333333333333333, t_2, abs(t_m)), Float64(k * k), Float64(t_2 * 2.0)) / Float64(l * l)) * Float64(k * k)));
	elseif (t_m <= 2e+123)
		tmp = Float64(Float64(l / Float64(Float64(t_m * t_m) * Float64(t_m * k))) * Float64(l / k));
	else
		tmp = Float64(Float64(l / Float64(Float64(k * t_m) * Float64(k * t_m))) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.6e-69], N[(2.0 / N[(N[(N[(N[(0.3333333333333333 * t$95$2 + N[Abs[t$95$m], $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2e+123], N[(N[(l / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(k * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.6000000000000002e-69

   1. Initial program 34.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 17.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\tan k \cdot \left(\mathsf{fma}\left(k, \frac{k}{t \cdot t}, 1\right) + 1\right)\right)}{\ell \cdot \ell}}} \]

   4. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

   6. Applied rewrites 58.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \left(t \cdot t\right) \cdot t, \left|t\right|\right), k \cdot k, \left(\left(t \cdot t\right) \cdot t\right) \cdot 2\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   if 2.6000000000000002e-69 < t < 1.99999999999999996e123

   1. Initial program 72.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 72.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\tan k \cdot \left(\mathsf{fma}\left(k, \frac{k}{t \cdot t}, 1\right) + 1\right)\right)}{\ell \cdot \ell}}} \]

   4. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      3. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      8. associate-*r/ N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      9. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \color{blue}{\ell} \]

      11. lower-*.f64 63.0

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \color{blue}{\ell} \]

   6. Applied rewrites 63.0%

      \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      5. pow3 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)} \cdot \ell \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)} \cdot \ell \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)} \cdot \ell \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)} \cdot \ell \]

      10. pow3 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      12. lift-*.f64 66.8

          \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 66.8%

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \color{blue}{\ell} \]

      2. lift-/.f64 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. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. associate-*l/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      9. times-frac N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]

   10. Applied rewrites 74.0%

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\frac{\ell}{k}} \]

   if 1.99999999999999996e123 < t

   1. Initial program 61.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      2. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. unpow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      8. unpow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      9. unpow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

      11. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      12. lower-*.f64 56.9

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   4. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      9. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      10. unpow3 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      13. times-frac N/A

          \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{2}} \cdot \color{blue}{\frac{\ell}{t}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{2}} \cdot \color{blue}{\frac{\ell}{t}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{2}} \cdot \frac{\color{blue}{\ell}}{t} \]

      16. pow2 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{2}} \cdot \frac{\ell}{t} \]

      17. pow2 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{t} \]

      18. unswap-sqr N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{t} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{t} \]

      20. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{t} \]

      21. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{t} \]

      22. lower-/.f64 88.3

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\color{blue}{t}} \]

   6. Applied rewrites 88.3%

      \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\frac{\ell}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 69.8% accurate, 4.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.4 \cdot 10^{-70}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\_m}\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{+123}:\\ \;\;\;\;\frac{\ell}{\left(t\_m \cdot t\_m\right) \cdot \left(t\_m \cdot k\right)} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(k \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.4e-70)
    (/ (* (* l l) 2.0) (* (* (* k k) (* k k)) t_m))
    (if (<= t_m 2e+123)
      (* (/ l (* (* t_m t_m) (* t_m k))) (/ l k))
      (* (/ l (* (* k t_m) (* k t_m))) (/ l t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.4e-70) {
		tmp = ((l * l) * 2.0) / (((k * k) * (k * k)) * t_m);
	} else if (t_m <= 2e+123) {
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k);
	} else {
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
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_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 5.4d-70) then
        tmp = ((l * l) * 2.0d0) / (((k * k) * (k * k)) * t_m)
    else if (t_m <= 2d+123) then
        tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k)
    else
        tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.4e-70) {
		tmp = ((l * l) * 2.0) / (((k * k) * (k * k)) * t_m);
	} else if (t_m <= 2e+123) {
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k);
	} else {
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 5.4e-70:
		tmp = ((l * l) * 2.0) / (((k * k) * (k * k)) * t_m)
	elif t_m <= 2e+123:
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k)
	else:
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.4e-70)
		tmp = Float64(Float64(Float64(l * l) * 2.0) / Float64(Float64(Float64(k * k) * Float64(k * k)) * t_m));
	elseif (t_m <= 2e+123)
		tmp = Float64(Float64(l / Float64(Float64(t_m * t_m) * Float64(t_m * k))) * Float64(l / k));
	else
		tmp = Float64(Float64(l / Float64(Float64(k * t_m) * Float64(k * t_m))) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 5.4e-70)
		tmp = ((l * l) * 2.0) / (((k * k) * (k * k)) * t_m);
	elseif (t_m <= 2e+123)
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k);
	else
		tmp = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.4e-70], N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2e+123], N[(N[(l / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(k * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.4000000000000003e-70

   1. Initial program 34.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 17.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\tan k \cdot \left(\mathsf{fma}\left(k, \frac{k}{t \cdot t}, 1\right) + 1\right)\right)}{\ell \cdot \ell}}} \]

   4. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

   6. Applied rewrites 69.9%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}} \]

   7. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

      3. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2} \cdot 2}{{k}^{4} \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot 2}{{k}^{4} \cdot t} \]

      5. pow2 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{{k}^{4} \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{{k}^{4} \cdot t} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{{k}^{4} \cdot t} \]

      8. sqr-pow N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{\left({k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{\left({k}^{2} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      12. pow2 N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

      14. pow2 N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

      15. lift-*.f64 55.9

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. Applied rewrites 55.9%

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   if 5.4000000000000003e-70 < t < 1.99999999999999996e123

   1. Initial program 71.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 72.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\tan k \cdot \left(\mathsf{fma}\left(k, \frac{k}{t \cdot t}, 1\right) + 1\right)\right)}{\ell \cdot \ell}}} \]

   4. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      3. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      8. associate-*r/ N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      9. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \color{blue}{\ell} \]

      11. lower-*.f64 63.0

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \color{blue}{\ell} \]

   6. Applied rewrites 63.0%

      \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      5. pow3 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)} \cdot \ell \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)} \cdot \ell \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)} \cdot \ell \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)} \cdot \ell \]

      10. pow3 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      12. lift-*.f64 66.7

          \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 66.7%

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \color{blue}{\ell} \]

      2. lift-/.f64 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. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. associate-*l/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      9. times-frac N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]

   10. Applied rewrites 73.9%

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\frac{\ell}{k}} \]

   if 1.99999999999999996e123 < t

   1. Initial program 61.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      2. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. unpow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      8. unpow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      9. unpow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

      11. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      12. lower-*.f64 56.9

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   4. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      9. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      10. unpow3 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      13. times-frac N/A

          \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{2}} \cdot \color{blue}{\frac{\ell}{t}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{2}} \cdot \color{blue}{\frac{\ell}{t}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{2}} \cdot \frac{\color{blue}{\ell}}{t} \]

      16. pow2 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{2}} \cdot \frac{\ell}{t} \]

      17. pow2 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{t} \]

      18. unswap-sqr N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{t} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{t} \]

      20. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{t} \]

      21. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{t} \]

      22. lower-/.f64 88.3

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\color{blue}{t}} \]

   6. Applied rewrites 88.3%

      \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\frac{\ell}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 69.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{\left(k \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)} \cdot \frac{\ell}{t\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.5 \cdot 10^{-152}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{+123}:\\ \;\;\;\;\frac{\ell}{\left(t\_m \cdot t\_m\right) \cdot \left(t\_m \cdot k\right)} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (/ l (* (* k t_m) (* k t_m))) (/ l t_m))))
   (*
    t_s
    (if (<= t_m 3.5e-152)
      t_2
      (if (<= t_m 2e+123) (* (/ l (* (* t_m t_m) (* t_m k))) (/ l k)) t_2)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	double tmp;
	if (t_m <= 3.5e-152) {
		tmp = t_2;
	} else if (t_m <= 2e+123) {
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k);
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
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_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (l / ((k * t_m) * (k * t_m))) * (l / t_m)
    if (t_m <= 3.5d-152) then
        tmp = t_2
    else if (t_m <= 2d+123) then
        tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k)
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	double tmp;
	if (t_m <= 3.5e-152) {
		tmp = t_2;
	} else if (t_m <= 2e+123) {
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k);
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = (l / ((k * t_m) * (k * t_m))) * (l / t_m)
	tmp = 0
	if t_m <= 3.5e-152:
		tmp = t_2
	elif t_m <= 2e+123:
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k)
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(l / Float64(Float64(k * t_m) * Float64(k * t_m))) * Float64(l / t_m))
	tmp = 0.0
	if (t_m <= 3.5e-152)
		tmp = t_2;
	elseif (t_m <= 2e+123)
		tmp = Float64(Float64(l / Float64(Float64(t_m * t_m) * Float64(t_m * k))) * Float64(l / k));
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	tmp = 0.0;
	if (t_m <= 3.5e-152)
		tmp = t_2;
	elseif (t_m <= 2e+123)
		tmp = (l / ((t_m * t_m) * (t_m * k))) * (l / k);
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(l / N[(N[(k * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.5e-152], t$95$2, If[LessEqual[t$95$m, 2e+123], N[(N[(l / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], t$95$2]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.5000000000000001e-152 or 1.99999999999999996e123 < t

   1. Initial program 46.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      2. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. unpow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      8. unpow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      9. unpow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

      11. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      12. lower-*.f64 52.4

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   4. Applied rewrites 52.4%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      9. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      10. unpow3 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      13. times-frac N/A

          \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{2}} \cdot \color{blue}{\frac{\ell}{t}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{2}} \cdot \color{blue}{\frac{\ell}{t}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{2}} \cdot \frac{\color{blue}{\ell}}{t} \]

      16. pow2 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{2}} \cdot \frac{\ell}{t} \]

      17. pow2 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{t} \]

      18. unswap-sqr N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{t} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{t} \]

      20. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{t} \]

      21. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{t} \]

      22. lower-/.f64 71.4

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\color{blue}{t}} \]

   6. Applied rewrites 71.4%

      \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

   if 3.5000000000000001e-152 < t < 1.99999999999999996e123

   1. Initial program 64.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 64.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\tan k \cdot \left(\mathsf{fma}\left(k, \frac{k}{t \cdot t}, 1\right) + 1\right)\right)}{\ell \cdot \ell}}} \]

   4. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      3. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      8. associate-*r/ N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      9. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \color{blue}{\ell} \]

      11. lower-*.f64 57.9

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \color{blue}{\ell} \]

   6. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      5. pow3 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)} \cdot \ell \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)} \cdot \ell \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)} \cdot \ell \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)} \cdot \ell \]

      10. pow3 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      12. lift-*.f64 60.1

          \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 60.1%

      \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \color{blue}{\ell} \]

      2. lift-/.f64 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. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

      7. associate-*l/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      9. times-frac N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k}} \]

   10. Applied rewrites 67.0%

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\frac{\ell}{k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 69.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{\left(k \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)} \cdot \frac{\ell}{t\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.5 \cdot 10^{-73}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 6.1 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{\left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (/ l (* (* k t_m) (* k t_m))) (/ l t_m))))
   (*
    t_s
    (if (<= t_m 5.5e-73)
      t_2
      (if (<= t_m 6.1e+102)
        (* (/ l k) (/ l (* (* (* t_m t_m) t_m) k)))
        t_2)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	double tmp;
	if (t_m <= 5.5e-73) {
		tmp = t_2;
	} else if (t_m <= 6.1e+102) {
		tmp = (l / k) * (l / (((t_m * t_m) * t_m) * k));
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
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_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (l / ((k * t_m) * (k * t_m))) * (l / t_m)
    if (t_m <= 5.5d-73) then
        tmp = t_2
    else if (t_m <= 6.1d+102) then
        tmp = (l / k) * (l / (((t_m * t_m) * t_m) * k))
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	double tmp;
	if (t_m <= 5.5e-73) {
		tmp = t_2;
	} else if (t_m <= 6.1e+102) {
		tmp = (l / k) * (l / (((t_m * t_m) * t_m) * k));
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = (l / ((k * t_m) * (k * t_m))) * (l / t_m)
	tmp = 0
	if t_m <= 5.5e-73:
		tmp = t_2
	elif t_m <= 6.1e+102:
		tmp = (l / k) * (l / (((t_m * t_m) * t_m) * k))
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(l / Float64(Float64(k * t_m) * Float64(k * t_m))) * Float64(l / t_m))
	tmp = 0.0
	if (t_m <= 5.5e-73)
		tmp = t_2;
	elseif (t_m <= 6.1e+102)
		tmp = Float64(Float64(l / k) * Float64(l / Float64(Float64(Float64(t_m * t_m) * t_m) * k)));
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (l / ((k * t_m) * (k * t_m))) * (l / t_m);
	tmp = 0.0;
	if (t_m <= 5.5e-73)
		tmp = t_2;
	elseif (t_m <= 6.1e+102)
		tmp = (l / k) * (l / (((t_m * t_m) * t_m) * k));
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(l / N[(N[(k * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.5e-73], t$95$2, If[LessEqual[t$95$m, 6.1e+102], N[(N[(l / k), $MachinePrecision] * N[(l / N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.50000000000000006e-73 or 6.1000000000000002e102 < t

   1. Initial program 46.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      2. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. unpow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      8. unpow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      9. unpow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

      11. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      12. lower-*.f64 50.8

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      9. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      10. unpow3 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      13. times-frac N/A

          \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{2}} \cdot \color{blue}{\frac{\ell}{t}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{2}} \cdot \color{blue}{\frac{\ell}{t}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{2}} \cdot \frac{\color{blue}{\ell}}{t} \]

      16. pow2 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{2}} \cdot \frac{\ell}{t} \]

      17. pow2 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{t} \]

      18. unswap-sqr N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{t} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{t} \]

      20. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{t} \]

      21. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{t} \]

      22. lower-/.f64 68.1

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\color{blue}{t}} \]

   6. Applied rewrites 68.1%

      \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

   if 5.50000000000000006e-73 < t < 6.1000000000000002e102

   1. Initial program 74.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      2. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. unpow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      8. unpow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      9. unpow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

      11. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      12. lower-*.f64 64.9

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   4. Applied rewrites 64.9%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      8. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      10. times-frac N/A

          \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\ell}{k \cdot {t}^{3}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\ell}{k \cdot {t}^{3}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\color{blue}{\ell}}{k \cdot {t}^{3}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell}{\color{blue}{k \cdot {t}^{3}}} \]

      14. *-commutative N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{k}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{k}} \]

      16. pow3 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \]

      18. lift-*.f64 73.1

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \]

   6. Applied rewrites 73.1%

      \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 67.3% accurate, 4.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{\left(\left(k \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)\right) \cdot t\_m} \cdot \ell\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.5 \cdot 10^{-73}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 7.6 \cdot 10^{+96}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{\left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (/ l (* (* (* k t_m) (* k t_m)) t_m)) l)))
   (*
    t_s
    (if (<= t_m 5.5e-73)
      t_2
      (if (<= t_m 7.6e+96) (* (/ l k) (/ l (* (* (* t_m t_m) t_m) k))) t_2)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (l / (((k * t_m) * (k * t_m)) * t_m)) * l;
	double tmp;
	if (t_m <= 5.5e-73) {
		tmp = t_2;
	} else if (t_m <= 7.6e+96) {
		tmp = (l / k) * (l / (((t_m * t_m) * t_m) * k));
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
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_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (l / (((k * t_m) * (k * t_m)) * t_m)) * l
    if (t_m <= 5.5d-73) then
        tmp = t_2
    else if (t_m <= 7.6d+96) then
        tmp = (l / k) * (l / (((t_m * t_m) * t_m) * k))
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (l / (((k * t_m) * (k * t_m)) * t_m)) * l;
	double tmp;
	if (t_m <= 5.5e-73) {
		tmp = t_2;
	} else if (t_m <= 7.6e+96) {
		tmp = (l / k) * (l / (((t_m * t_m) * t_m) * k));
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = (l / (((k * t_m) * (k * t_m)) * t_m)) * l
	tmp = 0
	if t_m <= 5.5e-73:
		tmp = t_2
	elif t_m <= 7.6e+96:
		tmp = (l / k) * (l / (((t_m * t_m) * t_m) * k))
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(l / Float64(Float64(Float64(k * t_m) * Float64(k * t_m)) * t_m)) * l)
	tmp = 0.0
	if (t_m <= 5.5e-73)
		tmp = t_2;
	elseif (t_m <= 7.6e+96)
		tmp = Float64(Float64(l / k) * Float64(l / Float64(Float64(Float64(t_m * t_m) * t_m) * k)));
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (l / (((k * t_m) * (k * t_m)) * t_m)) * l;
	tmp = 0.0;
	if (t_m <= 5.5e-73)
		tmp = t_2;
	elseif (t_m <= 7.6e+96)
		tmp = (l / k) * (l / (((t_m * t_m) * t_m) * k));
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(l / N[(N[(N[(k * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.5e-73], t$95$2, If[LessEqual[t$95$m, 7.6e+96], N[(N[(l / k), $MachinePrecision] * N[(l / N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.50000000000000006e-73 or 7.6000000000000003e96 < t

   1. Initial program 46.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 37.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\tan k \cdot \left(\mathsf{fma}\left(k, \frac{k}{t \cdot t}, 1\right) + 1\right)\right)}{\ell \cdot \ell}}} \]

   4. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      3. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      8. associate-*r/ N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      9. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \color{blue}{\ell} \]

      11. lower-*.f64 51.2

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \color{blue}{\ell} \]

   6. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      5. pow2 N/A

         \[\leadsto \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      6. pow3 N/A

         \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{3}} \cdot \ell \]

      7. unpow3 N/A

         \[\leadsto \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      8. pow2 N/A

         \[\leadsto \frac{\ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot t} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot t} \cdot \ell \]

      11. pow2 N/A

          \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \cdot \ell \]

      12. pow2 N/A

          \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \cdot \ell \]

      13. unswap-sqr N/A

          \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t} \cdot \ell \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t} \cdot \ell \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t} \cdot \ell \]

      16. lower-*.f64 65.3

          \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t} \cdot \ell \]

   8. Applied rewrites 65.3%

      \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t} \cdot \ell \]

   if 5.50000000000000006e-73 < t < 7.6000000000000003e96

   1. Initial program 74.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      2. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. unpow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      8. unpow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      9. unpow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

      11. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      12. lower-*.f64 64.5

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   4. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      8. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      10. times-frac N/A

          \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\ell}{k \cdot {t}^{3}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\ell}{k \cdot {t}^{3}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\color{blue}{\ell}}{k \cdot {t}^{3}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell}{\color{blue}{k \cdot {t}^{3}}} \]

      14. *-commutative N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{k}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{k}} \]

      16. pow3 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \]

      18. lift-*.f64 72.7

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k} \]

   6. Applied rewrites 72.7%

      \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 65.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\ell}{\left(\left(k \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)\right) \cdot t\_m} \cdot \ell\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (/ l (* (* (* k t_m) (* k t_m)) t_m)) l)))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / (((k * t_m) * (k * t_m)) * t_m)) * l);
}

Fortran

t\_m =     private
t\_s =     private
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_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l / (((k * t_m) * (k * t_m)) * t_m)) * l)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / (((k * t_m) * (k * t_m)) * t_m)) * l);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l / (((k * t_m) * (k * t_m)) * t_m)) * l)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l / Float64(Float64(Float64(k * t_m) * Float64(k * t_m)) * t_m)) * l))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l / (((k * t_m) * (k * t_m)) * t_m)) * l);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l / N[(N[(N[(k * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Applied rewrites 48.1%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\tan k \cdot \left(\mathsf{fma}\left(k, \frac{k}{t \cdot t}, 1\right) + 1\right)\right)}{\ell \cdot \ell}}} \]

4. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   2. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   3. pow3 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

   8. associate-*r/ N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

   9. lift-/.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

   10. *-commutative N/A

       \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \color{blue}{\ell} \]

   11. lower-*.f64 54.8

       \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \color{blue}{\ell} \]

6. Applied rewrites 54.8%

   \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

   5. pow2 N/A

      \[\leadsto \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

   6. pow3 N/A

      \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{3}} \cdot \ell \]

   7. unpow3 N/A

      \[\leadsto \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

   8. pow2 N/A

      \[\leadsto \frac{\ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \cdot \ell \]

   9. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot t} \cdot \ell \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot t} \cdot \ell \]

   11. pow2 N/A

       \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \cdot \ell \]

   12. pow2 N/A

       \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \cdot \ell \]

   13. unswap-sqr N/A

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t} \cdot \ell \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t} \cdot \ell \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t} \cdot \ell \]

   16. lower-*.f64 65.8

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t} \cdot \ell \]

8. Applied rewrites 65.8%

   \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t} \cdot \ell \]
9. Add Preprocessing

Alternative 16: 59.1% accurate, 6.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\ell}{k \cdot \left(\left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right) \cdot k\right)} \cdot \ell\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (/ l (* k (* (* (* t_m t_m) t_m) k))) l)))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / (k * (((t_m * t_m) * t_m) * k))) * l);
}

Fortran

t\_m =     private
t\_s =     private
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_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l / (k * (((t_m * t_m) * t_m) * k))) * l)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / (k * (((t_m * t_m) * t_m) * k))) * l);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l / (k * (((t_m * t_m) * t_m) * k))) * l)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l / Float64(k * Float64(Float64(Float64(t_m * t_m) * t_m) * k))) * l))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l / (k * (((t_m * t_m) * t_m) * k))) * l);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l / N[(k * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Applied rewrites 48.1%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right) \cdot \left(\tan k \cdot \left(\mathsf{fma}\left(k, \frac{k}{t \cdot t}, 1\right) + 1\right)\right)}{\ell \cdot \ell}}} \]

4. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   2. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   3. pow3 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

   8. associate-*r/ N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

   9. lift-/.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

   10. *-commutative N/A

       \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \color{blue}{\ell} \]

   11. lower-*.f64 54.8

       \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \color{blue}{\ell} \]

6. Applied rewrites 54.8%

   \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

   5. pow3 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}} \cdot \ell \]

   6. associate-*l* N/A

      \[\leadsto \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)} \cdot \ell \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)} \cdot \ell \]

   8. *-commutative N/A

      \[\leadsto \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)} \cdot \ell \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)} \cdot \ell \]

   10. pow3 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

   12. lift-*.f64 59.1

       \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]

8. Applied rewrites 59.1%

   \[\leadsto \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \cdot \ell \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025134 
(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))))